What are the correct version numbers for C#?

C# language version history:

These are the versions of C# known about at the time of this writing:

• C# 1.0 released with .NET 1.0 and VS2002 (January 2002)
• C# 1.2 (bizarrely enough); released with .NET 1.1 and VS2003 (April 2003). First version to call Dispose on IEnumerators which implemented IDisposable. A few other small features.
• C# 2.0 released with .NET 2.0 and VS2005 (November 2005). Major new features: generics, anonymous methods, nullable types, and iterator blocks
• C# 3.0 released with .NET 3.5 and VS2008 (November 2007). Major new features: lambda expressions, extension methods, expression trees, anonymous types, implicit typing (var), and query expressions
• C# 4.0 released with .NET 4 and VS2010 (April 2010). Major new features: late binding (dynamic), delegate and interface generic variance, more COM support, named arguments, tuple data type and optional parameters
• C# 5.0 released with .NET 4.5 and VS2012 (August 2012). Major features: async programming, and caller info attributes. Breaking change: loop variable closure.
• C# 6.0 released with .NET 4.6 and VS2015 (July 2015). Implemented by Roslyn. Features: initializers for automatically implemented properties, using directives to import static members, exception filters, element initializers, await in catch and finally, extension Add methods in collection initializers.
• C# 7.0 released with .NET 4.7 and VS2017 (March 2017). Major new features: tuples, ref locals and ref return, pattern matching (including pattern-based switch statements), inline out parameter declarations, local functions, binary literals, digit separators, and arbitrary async returns.
• C# 7.1 released with VS2017 v15.3 (August 2017). New features: async main, tuple member name inference, default expression, and pattern matching with generics.
• C# 7.2 released with VS2017 v15.5 (November 2017). New features: private protected access modifier, Span<T>, aka interior pointer, aka stackonly struct, and everything else.
• C# 7.3 released with VS2017 v15.7 (May 2018). New features: enum, delegate and unmanaged generic type constraints. ref reassignment. Unsafe improvements: stackalloc initialization, unpinned indexed fixed buffers, custom fixed statements. Improved overloading resolution. Expression variables in initializers and queries. == and != defined for tuples. Auto-properties' backing fields can now be targeted by attributes.
• C# 8.0 released with .NET Core 3.0 and VS2019 v16.3 (September 2019). Major new features: nullable reference-types, asynchronous streams, indices and ranges, readonly members, using declarations, default interface methods, static local functions, and enhancement of interpolated verbatim strings.
• C# 9.0 released with .NET 5.0 and VS2019 v16.8 (November 2020). Major new features: init-only properties, records, with-expressions, data classes, positional records, top-level programs, improved pattern matching (simple type patterns, relational patterns, logical patterns), improved target typing (target-type new expressions, target typed ?? and ?), and covariant returns. Minor features: relax ordering of ref and partial modifiers, parameter null checking, lambda discard parameters, native ints, attributes on local functions, function pointers, static lambdas, extension GetEnumerator, module initializers, and extending partial.
• C# 10.0 released with .NET 6.0 (November 2021). Major new features: record structs, struct parameterless constructors, interpolated string handlers, global using directives, file-scoped namespace declarations, extended property patterns, const interpolated strings, mixed assignment and declaration in deconstruction, async method builders (via attributes) for individual methods, the CallerArgumentExpression attribute for parameters, enhanced #line pragmas.

In response to the OP's question:

What are the correct version numbers for C#? What came out when? Why can't I find any answers about C# 3.5?

There is no such thing as C# 3.5 - the cause of confusion here is that the C# 3.0 is present in .NET 3.5. The language and framework are versioned independently, however - as is the CLR, which is at version 2.0 for .NET 2.0 through 3.5, .NET 4 introducing CLR 4.0, service packs notwithstanding. The CLR in .NET 4.5 has various improvements, but the versioning is unclear: in some places it may be referred to as CLR 4.5 (this MSDN page used to refer to it that way, for example), but the Environment.Version property still reports 4.0.xxx.

As of May 3, 2017, the C# Language Team created a history of C# versions and features on their GitHub repository: Features Added in C# Language Versions. There is also a page that tracks upcoming and recently implemented language features.

This is the same as most answers here, but tabularized for ease, and it has Visual Studio and .NET versions for completeness.

Versions since .NET Core

* - .NET 5.0 is not a newer version of .NET framework but .NET Core 3. Starting from .NET 5.0, there are no newer versions of .NET full framework.

** - There are no separate CLR (CoreCLR) versions for .NET Core. Whatever is the .NET Core version is the CoreCLR version. So not mentioning it.

Note: .NET development is pretty much independent of VS these days, there is no correlation between versions of each.
Refer to ".NET Framework versions and dependencies" and ".NET release cadence" for more.

The biggest problem when dealing with C#'s version numbers is the fact that it is not tied to a version of the .NET Framework, which it appears to be due to the synchronized releases between Visual Studio and the .NET Framework.

The version of C# is actually bound to the compiler, not the framework. For instance, in Visual Studio 2008 you can write C# 3.0 and target .NET Framework 2.0, 3.0 and 3.5. The C# 3.0 nomenclature describes the version of the code syntax and supported features in the same way that ANSI C89, C90, and C99 describe the code syntax and features for C.

Take a look at Mono, and you will see that Mono 2.0 (mostly implemented version 2.0 of the .NET Framework from the ECMA specifications) supports the C# 3.0 syntax and features.