Dmitry Sotnikov explains how to install .NET and PowerShell on Windows Server 2008 in its Server Core configuration. It is necessary to tweak the .NET setup with Orca, a low-level editor for Microsoft Installer files. Note this is unsupported.
The lack of PowerShell is an annoyance; the lack of .NET is a major obstacle to making use of Server Core, so this is interesting work. Sotnikov does not say whether ASP.NET springs to life; I presume it may be possible.
I imagine that one of the issues with .NET on Server Core is that some parts of the Framework will not work because dependencies are missing. Server Core has little in the way of a GUI, so I would not expect System.Windows.Forms or Windows Presentation Foundation to work; yet the .NET runtime is all or nothing. This is changing; Microsoft has announced a Client Profile Setup to reduce the runtime size in .NET 3.5 SP1, for client applications.
What we now need is a Server Profile, tailored to work on Server Core.