I used https://github.com/lelandrichardson/enzyme-example-karma-webpack for a default setup and then put the following in the test file.

test.js

import React, { Component } from 'react';
import styled from 'styled-components';
import { mount } from 'enzyme';

describe("A suite", function() {
  it("should select an element using standard and non-standard DOM attributes", function() {
    const StyledComponent = styled.div`
      background-color: black;
    `;
    class Test extends Component {
      render () {
        return (
          <div>
            <StyledComponent value="test" />
            <StyledComponent value="test" nonStandard="test" />
          </div>
        );
      }
    }

    expect(mount(<Test />).find({
      value: 'test'
    }).length).toBe(2, 'standard');

    expect(mount(<Test />).find({
      nonStandard: 'test'
    }).length).toBe(1, 'non-standard');

    expect(mount(<Test />).find({
      nonStandard: 'test',
      value: 'test'
    }).length).toBe(1, 'mixture');
  });
});

The purpose of these is to use them in the tagged literal as props, so need non string values to work. Don't have the capacity to try at the moment if that will yield expected results but will have a go on Monday. Thanks again for your response, @mxstbr!