Omnisharp[1] is build on top of Roslyn so as for now it supports only C# / VB. As far as I know they are working on adding plugin support for it for non-Roslyn language so maybe in the future we will have F# plugin there ( actually, as far as I know, that's MSFT plan for adding F# support in VS Code)
F# has had opensourced compiler[2] (written in F#) before it was cool in .Net stack and for most tooling things we are using FSharp.Compiler.Services[3] project - which basically provides all features of F# compiler (and much more) as normal .Net library. Projects like Visual F# Power Tools[4] and Xamarin Studio F# support[5] are using it to provide very interesting F# IDE features.
Since Vim, Emacs, Sublime and Atom (even though Ionide is written in F# and compiled to JS we can't use all .Net libraries) don't have direct access to .Net assemblies, community have created FsAutoComplete[6] project which is just command line wrapper around FSharp.Compiler.Services and works as "standard I/O server" where we send requests (for example if user wants completion list) and get appropriate responses.
> for now it supports only C# / VB
Nope, there is not any support for VB at this point in time. It's C# only I'm afraid!
As cloudroutine mentions, the older omnisharp-server implementation is NRefactory based, whilst omnisharp-roslyn (as the name implies) has been built atop roslyn.
omnisharp-roslyn retains a bit of NRefactory (the code actions part[1]), which was called NR6Pack and has now been renamed as Refactoring Essentials [2].
Anyway, there's been a lot of recent changes in the works, as someone has already mentioned, omnisharp-rolsyn currently has an open PR [3] which will provide a plugin system to omnisharp-roslyn. Once introduced, it will mean that the various omnisharp editors/clients can pick and choose which plugins they wish to bundle with their editor. This may include F#, VB, Cake, Fake and many others.
F# has had opensourced compiler[2] (written in F#) before it was cool in .Net stack and for most tooling things we are using FSharp.Compiler.Services[3] project - which basically provides all features of F# compiler (and much more) as normal .Net library. Projects like Visual F# Power Tools[4] and Xamarin Studio F# support[5] are using it to provide very interesting F# IDE features.
Since Vim, Emacs, Sublime and Atom (even though Ionide is written in F# and compiled to JS we can't use all .Net libraries) don't have direct access to .Net assemblies, community have created FsAutoComplete[6] project which is just command line wrapper around FSharp.Compiler.Services and works as "standard I/O server" where we send requests (for example if user wants completion list) and get appropriate responses.
[1] http://www.omnisharp.net/
[2] https://github.com/Microsoft/visualfsharp
[3] https://github.com/fsharp/FSharp.Compiler.Service
[4] https://github.com/fsprojects/VisualFSharpPowerTools
[5] https://github.com/fsharp/xamarin-monodevelop-fsharp-addin
[6] https://github.com/fsharp/FsAutoComplete