How do I install dotNet Framework 3.51 on Windows Server 2012 core?

You can use Powershell to show the available Windows Features:

Import-Module ServerManager
Get-WindowsFeature | out-file C:\WindowsFeatures.txt -width 300

Because the data is a long and wide it is better to pipe it into a file and review it there.

it shows that '.NET Framework 3.5 Features' are available, but '.NET Framework 3.5 (includes .NET 2.0 and 3.0)' has been removed.

However, the files are still on the DVD, you can use:

dism /online /enable-feature /featurename:NetFX3 /source:D:\sources\sxs /all /LimitAccess

or the PowerShell 'Add-WindowsFeature' commandlet to install it:

Add-WindowsFeature –name NET-Framework-Core -source D:\sources\sxs

Edit: On Server 2012 R2 with Update 1 installed (KB2919355) this doesn't work if you are using the original DVD/ISO. You need a DVD or ISO which includes that update. For new installations, install .NET 3.5 before you apply KB2919355.

dism /online /enable-feature /featurename:NetFx3 /All /LimitAccess /Source:x:\sources\sxs

Where x: is the drive letter of the installation media or mapped network share that contains a copy of the installation files.

A Windows update may be causing the problem: KB2966827 i.e. An update for 3.5 was installed even though the feature wasn't installed

Read here at the very bottom in the comments: http://social.technet.microsoft.com/Forums/windowsserver/en-US/5c16b88a-0f19-4aea-ad65-38f0bdb59b9c/install-net-framework-35-on-windows-server-2012-behind-the-firewall-does-not-recognize-sources?forum=winserver8gen