CompareString.asm

<Main>g__Operator|4_0 (String, String)
L0000	jmp	System.String.Equals(System.String, System.String)

<Main>g__Equals|4_1 (String, String)
L0000	mov	rax, 0x7fff58527b68
L000a	mov	rax, [rax]
L000d	cmp	[rcx], ecx
L000f	jmp	rax

<Main>g__EqualsComparer|4_2<T> (T, T)
 ?
 
 <Main>g__EqualsComparerString|4_3 (String, String)
L0000	mov	r8, rcx
L0003	mov	rax, rdx
L0006	mov	rcx, 0x1cfdfc42c78
L0010	mov	rcx, [rcx]
L0013	mov	rdx, r8
L0016	mov	r8, rax
L0019	mov	rax, 0x7fff58534f50
L0023	mov	rax, [rax]
L0026	jmp	rax

<Main>g__ObjOperator|4_4 (Object, Object)
L0000	cmp	rcx, rdx
L0003	sete	al
L0006	movzx	eax, al
L0009	ret	

<Main>g__ObjEquals|4_5 (Object, Object)
L0000	mov	rax, [rcx]
L0003	mov	rax, [rax+0x40]
L0007	mov	rax, [rax+0x10]
L000b	jmp	rax

<Main>g__ObjReferenceEquals|4_6 (String, String)
L0000	cmp	rcx, rdx
L0003	sete	al
L0006	movzx	eax, al
L0009	ret

<Main>g__OperatorT|4_7<T> (T, T)
 ?
 
 EmbeddedAttribute..ctor
L0000	ret	

CompareString.cs

foreach (var (s1, s2) in new[] { ("s", "s"), (new String("s"), new String("s")) })
{
    "========================================================".Dump();
    Operator(s1, s2).Dump("string ==");
    Equals(s1, s2).Dump("string Equals");
    EqualsComparer(s1, s2).Dump("T EqualityComparer");
    EqualsComparerString(s1, s2).Dump("string EqualityComparer");

    ObjOperator(s1, s2).Dump("obj ==");
    ObjEquals(s1, s2).Dump("object Equals");
    ObjReferenceEquals(s1, s2).Dump("ReferenceEquals");
    OperatorT(s1, s2).Dump("T ==");
}

bool Operator(string l, string r) => l == r;
bool Equals(string l, string r) => l.Equals(r);
bool EqualsComparer<T>(T l, T r) => EqualityComparer<T>.Default.Equals(l, r);
bool EqualsComparerString(string l, string r) => EqualityComparer<string>.Default.Equals(l, r);

bool ObjOperator(object l, object r) => l == r;
bool ObjEquals(object l, object r) => l.Equals(r);
bool ObjReferenceEquals(string l, string r) => Object.ReferenceEquals(l, r);
bool OperatorT<T>(T l, T r) where T: class => l == r;

Result.txt

========================================================
string ==
True
string Equals
True
T EqualityComparer
True
string EqualityComparer
True
obj ==
True
object Equals
True
ReferenceEquals
True
T ==
True
========================================================
string ==
True
string Equals
True
T EqualityComparer
True
string EqualityComparer
True
obj ==
False
object Equals
True
ReferenceEquals
False
T ==
False

StringCreate.asm

L0000	sub	rsp, 0x38 // only this line is `var s1 = "s";`
L0004	xor	eax, eax  // below are all `var s2 = new String("s");`
L0006	mov	[rsp+0x28], rax
L000b	mov	rdx, 0x1ad70eb3010
L0015	mov	rdx, [rdx]
L0018	test	rdx, rdx
L001b	jne	short L0023
L001d	xor	ecx, ecx
L001f	xor	eax, eax
L0021	jmp	short L0033
L0023	cmp	[rdx], edx
L0025	cmp	[rdx], edx
L0027	add	rdx, 0xc
L002b	mov	rcx, rdx
L002e	mov	eax, 1
L0033	lea	rdx, [rsp+0x28]
L0038	mov	[rdx], rcx
L003b	mov	[rdx+8], eax
L003e	lea	rdx, [rsp+0x28]
L0043	xor	ecx, ecx
L0045	call	System.String..ctor(System.ReadOnlySpan`1<Char>)
L004a	nop	
L004b	add	rsp, 0x38
L004f	ret	

StringCreate.cs

var s1 = "s";
var s2 = new String("s");

StringCreate.il

IL_0000	ldstr	"s"
IL_0005	call	String.op_Implicit (String)
IL_000A	newobj	String..ctor
IL_000F	pop	
IL_0010	ret