I can find a lot of obsolete threads but a lot have been changed the last months.
Is there currently any possibility to run .NET Core and/or ASP.NET Core on Raspberr
.NET Core is scheduled to run on ARM based OS like Raspian / Windows 10 IoT for Q4 2016 / Q1 2017. https://blogs.msdn.microsoft.com/dotnet/2016/07/15/net-core-roadmap/
At this point of time I think you can follow the installations for Debian but you should get a exec format error
IIRC.
Edit 3/10/17: first official instruction is online now: https://github.com/dotnet/core/blob/master/samples/RaspberryPiInstructions.md
Officially for the 1.0 release, no. The only runtimes available for ARM is currently the Windows ones.
Also, Homebrew is for OSX. What you want to be looking at in the future for on Raspbian is the Debian instructions.
Now you CAN run dotnet core applications on Raspberry PI 3, with no Mono installed at all.
I managed to run complex app on headless Ubuntu Server 16.04 and on Ubuntu Mate 16.04.
All you have to do is to get the build output of cross compilation of coreclr and corefx, remove private and pre-compiled dlls, and put your managed dlls on the PI along with them.
Then you can run it using "corerun" executable.
(I posted a video on that one but it seems to be illegal to post it here)
Anyhow, to answer your question : it is now possible to run dotnet core apps on RPI3, although it isn't released yet.
Yes, .Net Core will run on Raspberry Pi 3.
You can find the procedure for installation from here in stackoverflow.
In more details you can find the below link also
http://reddyinfosoft.blogspot.com/2020/01/how-to-install-net-core-31-on-raspberry.html
You can get a (seemingly) stable build of .Net Core on Raspberry Pi 2 or 3 by following Peter Marcu's instructions here: https://github.com/dotnet/core/blob/master/samples/RaspberryPiInstructions.md
You don't need to use "corerun", you can use "dotnet". You build on your dev machine and copy to your Pi.
Update: I realise now the original question specifically mentioned Rasbian (Jessie). The method at that link requires Ubuntu 14.04 or 16.04. But if that isn't a problem I think you should be good to go.