This is a fork of Chip Bennets idea to implement breadcrumbs into WP Core. The code is procedural and really not the best - but it would fit into core much easier this way as when implementing it with real OOP patterns.

breadcrumb.navigation.php

<?php
/**
 * Add navigation breadcrumb
 */
 
function wp_nav_breadcrumb() {
	// Separator between breadcrumb links
	$delimiter = apply_filters( 'wp_nav_breadcrumb_delimiter', ' &raquo; ' );	
	// Show taxonomy crumbs for single posts
	$show_tax_crumb_for_single = apply_filters( 'wp_nav_breadcrumb_show_tax_crumb_for_single', true );

	$pagination = false;
	
	// Base link
	$baselink = sprintf(
		'<a href="%s">%s</a>',
		home_url( '/' ),
		apply_filters( 'wp_nav_breadcrumb_home_text', __( 'Home' ) )
	);
	$hierarchy = $delimiter;
	
	// Base link	
	if ( is_front_page() ) {
		$context = wp_nav_breadcrumb_front_page( $base_link );
	} 
	
	// Define current location for 404 Error page
	if ( is_404() ) { 
		$context = wp_nav_breadcrumb_404();
	} 
	
	// Define current location for Search Results page
	elseif ( is_search() ) { 
		$context = wp_nav_breadcrumb_search( $delimiter );  
	} 
	
	// Define current location for the blog posts index
	elseif ( is_home() && ( 'page' == get_option( 'show_on_front' ) ) ) {
		$context = wp_nav_breadcrumb_blog();
	}
	
	// Define current location for archive pages
	elseif( is_archive() ) {
	
	    // Define Category Hierarchy Crumbs for Category Archive
		if ( is_category() ) { 
			$context = wp_nav_breadcrumb_category( $delimiter );
		} 		
			
		// Define current location for Tag Archives
		elseif ( is_tag() ) {	  
			$context = wp_nav_breadcrumb_tag( $delimiter );  
		} 
		
		// Define current location for Post Format Archives
		elseif ( get_post_format() && ! is_home() ) { 
			$context = wp_nav_breadcrumb_post_format( $delimiter );
		}
		
		// Define current location for Custom Taxonomy Archives
		elseif ( is_tax() ) {
			$context = wp_nav_breadcrumb_tax( $delimiter );
		}
		
		// Define current location for Author Archives
		elseif ( is_author() ) { 
			$context = wp_nav_breadcrumb_author( $delimiter );
		} 
	
		// Define Crumbs for Day/Year/Month Date-based Archives
		elseif ( is_date() ) { 
			$context = wp_nav_breadcrumb_date( $delimiter );
		} 
		
		// Define current location for Custom Post Type Archives
		elseif ( is_post_type_archive( get_post_type() ) ) {
			$context = wp_nav_breadcrumb_post_type_archive( $delimiter );
		}		
 
		// Define pagination for paged Archive pages
		if ( get_query_var( 'paged' ) ) {
			$pagination  = $delimiter;
			$pagination .= apply_filters( 'wp_nav_breadcrumb_paged_archive_text', sprintf( 'Page %s', get_query_var( 'paged' ) ) );
		}
	
	}
	
	// Define current location for singular pages
	elseif ( is_singular() ) {
	
		if ( is_page() && ! is_front_page() ) {
			$context = wp_nav_breadcrumb_page( $delimiter, $show_tax_crumb_for_single );
		}
	
		// Define Category and Parent Post Crumbs for Post Attachments
		elseif ( is_attachment() ) { 
			$context = wp_nav_breadcrumb_attachment( $delimiter, $show_tax_crumb_for_single );
		}
		
		// Define Category Hierarchy Crumbs for Single Posts
		elseif ( is_singular( 'post' ) ) { 
			$context = wp_nav_breadcrumb_single_blog_post( $delimiter, $show_tax_crumb_for_single );
		}
		
		// Define current location for Custom Post Types
		elseif ( is_singular( get_post_type() ) ) {
			$context = wp_nav_breadcrumb_single_post_type( $delimiter, $show_tax_crumb_for_single );
		}
 
		// Define pagination for singular pages
		if ( get_query_var( 'page' ) ) {
			$pagination  = $delimiter;
			$pagination .= apply_filters( 'wp_nav_breadcrumb_paged_post_text', sprintf( 'Page %s', get_query_var( 'page' ) ) );
		}
	
	}

	// Breadcrumb array
	$breadcrumb = array();
	
	// Opening element that wraps entire breadcrumb output
	$breadcrumb[] = apply_filters( 'wp_nav_breadcrumb_container_before', '<div id="breadcrumbs">' );
	// Opening breadcrumb element
	$breadcrumb[] = apply_filters( 'wp_nav_breadcrumb_container_open', '<nav class="crumbs">' );
	
	if ( $baselink ) {
		$breadcrumb[] = $baselink;
	}
	
	if ( $context['hierarchy'] ) {
		$breadcrumb[] = $context['hierarchy'];	
	}
	
	// Opening element that wraps the current page breadcrumb link
	$breadcrumb[] = apply_filters( 'wp_nav_breadcrumb_current_before', '<strong>' );
	// Current location
	$breadcrumb[] = $context['current_location'];
	//Closing element that wraps the current page breadcrumb link
	$breadcrumb[] = apply_filters( 'wp_nav_breadcrumb_current_after', '</strong>' );
	
	if ( $pagination ) {
		$breadcrumb[] = $pagination;
	}
	
	// Closing element that wraps entire breadcrumb output
	$breadcrumb[] = apply_filters( 'wp_nav_breadcrumb_container_after', '</div>' );
	// Closing breadcrumb element
	$breadcrumb[] = apply_filters( 'wp_nav_breadcrumb_container_close', '</nav>' );
	

	// Output the result
	echo implode( '', apply_filters( 'wp_nav_breadcrumb', $breadcrumb ) );
}

/**
 * Site Front Page navigation breadcrumb
 */
function wp_nav_breadcrumb_front_page( $base_link ) {
	return array(
		'hierarchy'        => false,
		'current_location' => $base_link
	);
}

/**
 * 404 navigation breadcrumb
 */
function wp_nav_breadcrumb_404() {
	return array(
		'hierarchy'        => false,
		'current_location' => apply_filters( 'wp_nav_breadcrumb_404_text', __( 'Page Not Found' ) )
	);
}

/**
 * Search results navigation breadcrumb
 */
function wp_nav_breadcrumb_search( $delimiter ) {
	$hierarchy  = apply_filters( 'wp_nav_breadcrumb_search_text', __( 'Search Results:' ) );
	$hierarchy .= $delimiter;

	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => get_search_query()
	);
}

/**
 * Blog posts index navigation breadcrumb
 */
function wp_nav_breadcrumb_blog( $blog_text ) {
	// Text for the 'Blog' link
	$blog_text = apply_filters( 'wp_nav_breadcrumb_blog_text', __( 'Blog' ) );
	if ( is_paged() ) {
		$blog_text = sprintf(
			'<a href="%s">%s</a>',
			get_permalink( get_option( 'page_for_posts' ) ),
			$blog_text
		);
	}
	
	return array(
		'hierarchy'        => false,
		'current_location' => $blog_text
	);
}

/**
 * Category archive index navigation breadcrumb
 */
function wp_nav_breadcrumb_category( $delimiter ) {
	global $wp_query;

	$cat_obj = $wp_query->get_queried_object();
	$current_cat = $cat_obj->term_id;
	$current_cat = get_category( $thisCat );
	$hierarchy  = apply_filters( 'wp_nav_breadbrumb_category_archive_text', __( 'Category Archive' ) );
	$hierarchy .= $delimiter;
	if ( $thisCat->parent != 0 ) {
		$hierarchy .= get_category_parents( get_category( $current_cat->parent ), true, $delimiter );
	} 
	
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => single_cat_title( '' , false )
	);
}

/**
 * Tag archive index navigation breadcrumb
 */
function wp_nav_breadcrumb_tag( $delimiter ) {			
	$hierarchy  = apply_filters( 'wp_nav_breadcrumb_tag_archive_text', __( 'Tag Archive' ) );
	$hierarchy .= $delimiter;

	return array(
		'hierarchy'        => ,
		'current_location' => single_tag_title( '' , false )
	);
}

/**
 * Post Format archive index navigation breadcrumb
 */
function wp_nav_breadcrumb_post_format( $delimiter ) {
	$hierarchy  = apply_filters( 'wp_nav_breadcrumb_post_format_archive_text', __( 'Post Format Archive' ) );
	$hierarchy .= $delmiter;
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => get_post_format_string( get_post_format() )
	);
}

/**
 * Custom Taxonomy archive index navigation breadcrumb
 */
function wp_nav_breadcrumb_tax( $delimiter ) {
	global $wp_query;

	$post_type_object = get_post_type_object( get_post_type() );
	$post_type_name   = $post_type_object->labels->name;
	$post_type_slug   = $post_type_object->name;

	$custom_tax = $wp_query->query_vars['taxonomy'];
	$custom_tax_object = get_taxonomy( $custom_tax );

	$hierarchy  = sprintf(
		'<a href="%s">%s</a>',
		get_post_type_archive_link( $post_type_slug ),
		$post_type_name
	);
	$hierarchy .= $delimiter;
	$hierarchy .= apply_filters( 'wp_nav_breadcrumb_taxonomy_archive_text', __( 'Archive Index for' ) );
	$hierarchy .= $delimiter;
	
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => single_term_title( '', false );
	);
}

/**
 * Author archive index navigation breadcrumb
 */
function wp_nav_breadcrumb_author( $delimiter ) {
	$hierarchy  = apply_filters( 'wp_nav_breadcrumb_author_archive_text', __( 'Posts Written by' ) );
	$hierarchy .= $delimiter;
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => get_the_author_meta( 'display_name', get_query_var( 'author' ) )
	);
}

/**
 * Date archive index navigation breadcrumb
 */
function wp_nav_breadcrumb_date( $delimiter ) {
	// Define Year/Month Hierarchy Crumbs for Day Archive
	if  ( is_day() ) {
		$date_string  = sprintf(
			'<a href="%s">%s</a> ',
			get_year_link( get_the_time( 'Y' ) ),
			get_the_time( 'Y' )
		);
		$date_string .= $delimiter . ' ';
		$date_string .= sprintf(
			'<a href="%s">%s</a> ',
			get_month_link( get_the_time( 'Y' ), get_the_time( 'm' ) ),
			get_the_time( 'F' )
		);
		$date_string .= $delimiter . ' ';
		$current_location = get_the_time('d'); 
	} 
	// Define Year Hierarchy Crumb for Month Archive
	elseif ( is_month() ) {
		$date_string  = sprintf(
			'<a href="%s">%s</a> ',
			get_year_link( get_the_time( 'Y' ) ),
			get_the_time( 'Y' )
		);
		$date_string .= $delimiter . ' ';
		$current_location = get_the_time( 'F' ); 
	} 
	// Set CurrentLocation for Year Archive
	elseif ( is_year() ) {
		$date_string = '';
		$current_location = get_the_time( 'Y' ); 
	}

	$hierarchy = sprintf( __( 'Posts Published in: %s' ), $date_string );

	return array(
		'hierarchy'        => apply_filters( 'wp_nav_breadcrumb_date_archive_text', $hierarchy ),
		'current_location' => $current_location
	);
}

/**
 * Custom post type archive index breadcrumb
 */
function wp_nav_breadcrumb_post_type_archive( $delimiter ) {

	$post_type_object = get_post_type_object( get_post_type() );
	$post_type_name   = $post_type_object->labels->name;

	$hierarchy  = apply_filters( 'wp_nav_breadcrumb_post_type_archive_text', __( 'Archive Index for' ) );
	$hierarchy .= $delimiter;

	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => $post_type_name
	);
}

/**
 * Static page navigation breadcrumb
 */
function wp_nav_breadcrumb_page( $delimiter ) {
	global $post;

	// Define Parent Page Hierarchy Crumbs for Hierarchical Pages
	if ( $post->post_parent ) { 
		$parent_id  = $post->post_parent;
		$pagecrumbs = array();
		while ( $parent_id ) {
			$page = get_page( $parent_id );
			$pagecrumbs[] = sprintf(
				'<a href="%s">%s</a>',
				get_permalink( $page->ID ),
				get_the_title( $page->ID )
			);
			$parent_id  = $page->post_parent;
		}
		$pagecrumbs = array_reverse( $pagecrumbs );
		foreach ( $pagecrumbs as $crumb ) {
			$hierarchy .= $crumb;
			$hierarchy .= $delimiter;
		}
	} 
		
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => get_the_title()
	);
}

/**
 * Attachment page navigation breadcrumb
 */
function wp_nav_breadcrumb_attachment( $delimiter, $show_tax_crumb_for_single ) {
	global $post;

	$hierarchy = '';
	// Get tax crumb
	if ( $show_tax_crumb_for_single ) {
		$cat_parents = '';
		$parent = get_post( $post->post_parent );
		if ( get_the_category( $parent->ID ) ) {
			$cat = get_the_category( $parent->ID ); 
			$cat_parents = get_category_parents( $cat[0], TRUE, $delimiter );
		}
		$hierarchy .= $cat_parents;
	} 
	// Get parent post
	$hierarchy .= sprintf(
		'<a href="%s">%s</a> ',
		get_permalink( $parent ),
		$parent->post_title
	);
	$hierarchy .= $delimiter;
	
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => get_the_title()
	);
}

/**
 * Single blog post navigation breadcrumb
 */
function wp_nav_breadcrumb_single_blog_post( $delimiter, $show_tax_crumb_for_single ) {
	$hierarchy = '';
	if ( $show_tax_crumb_for_single ) {
		$cats = get_the_category(); 
		// Assume the first category is current
		$current_cat = ( $cats ? $cats[0] : '' );
		// Determine if category is hierarchical
		$cat_is_hierarchical = false;
		foreach ( $cats as $cat ) {
			if ( '0' != $cat->parent ) {
				$cat_is_hierarchical = true;
				break;
			}
		}
		// If category is hierarchical,
		// ensure we have the correct child category
		if ( $cat_is_hierarchical ) {
			foreach ( $cats as $cat ) {
				$children = get_categories( array( 'parent' => $cat->term_id ) );
				if ( 0 == count( $children ) ) {
					$current_cat = $cat;
					break;
				}
			}
		}
		// Get the hierarchical list of category links
		$hierarchy .= get_category_parents( $current_cat, TRUE, $delimiter );
	}
	
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => get_the_title()
	);
}

/**
 * Single custom post navigation breadcrumb
 */
function wp_nav_breadcrumb_single_post_type( $delimiter, $show_tax_crumb_for_single ) {
	
	$hierarchy = '';
	$post_type_object = get_post_type_object( get_post_type() );
	$post_type_name   = $post_type_object->labels->name;
	$post_type_slug   = $post_type_object->name;
	
	if ( $show_tax_crumb_for_single ) {
		$taxonomies = get_object_taxonomies( get_post_type() );
		$taxonomy = ( ! empty( $taxonomies ) ? $taxonomies[0] : false );
		$terms = ( $taxonomy ? get_the_term_list( $post->ID, $taxonomy ) : false );
		if ( $terms ) {
			$hierarchy .= $terms;
			$hierarchy .= $delimiter;
		}
	}
	$hierarchy .= sprintf(
		'<a href="%s">%s</a>',
		get_post_type_archive_link( $post_type_slug ),
		$post_type_name
	);
			
	return array(
		'hierarchy'        => $hierarchy,
		'current_location' => get_the_title()
	);
}

changelog.md

1. consistent naming scheme for functions: wp_nav_menu() > wp_nav_breadcrumbs_*()
2. array() alignment for better readability on returns. fixed a typo with a missing $hierarchy value for wp_nav_breadcrumb_single_post_type() and removed typo with ; semicolon in the hierarchy return array of wp_nav_breadcrumb_author()
3. Added _nav to all filter names so they don't differentiate too much from the function names - easier to remember.
4. Added another forgotten _nav for the delimiter filter name.
5. Removed massive concatenating of strings. Reduced to bit-by-bit concatenation: One argument per line. Fixed an errornous semicolon on line 202. Changed CamelCased vars like $currentLocation to $current_location. Removed some unnecessary line breaks and moved global always to the beginning of a function. Removed oenology textdomain from ~334
6. Fixed a comparisson error on L#91: & vs. &&
7. Removed another camelCase as well as lowercased all booleans.