Possible causes of "Failed to initialize the Common Language Runtime (CLR)" when using SQL CLR

When using SQL CLR stored procedures, user defined functions, user defined data types, or user defined aggregates, you many notice one or more of the following error messages being logged in the SQL server error log when running on a 32 bit instance of either SQL Server 2005 or 2008 

Error message 1

Failed to initialize the Common Language Runtime (CLR) v2.0.50727 with HRESULT 0x8007000e.
You may fix the problem and try again later.

Error message 2

Msg 6511, Level 16, State 20, Line 1
Failed to initialize the Common Language Runtime (CLR) v2.0.50727 with HRESULT 0x80131022.
You may fix the problem and try again later.

Error message 3

Msg 6512, Level 16, State 27, Line 2
Failed to initialize the Common Language Runtime (CLR) v2.0.50727 with HRESULT 0x80004005.
You need to restart SQL server to use CLR integration features.

Error message 4

Msg 6513, Level 16, State 27, Line 1
Failed to initialize the Common Language Runtime (CLR) v2.0.50727 due to memory pressure.
Please restart SQL server in Address Windowing Extensions (AWE) mode to use CLR integration
features.

Error message 5

6513: Failed to initialize the Common Language Runtime (CLR) v2.0.50727 due to memory pressure.
This is probably due to memory pressure in the MemToLeave region of memory.
For more information, see the CLR integration documentation in SQL Server Books Online.

Error message 6

Failed to initialize the Common Language Runtime (CLR) v2.0.50727 with HRESULT 0x80131522.
You need to restart SQL server to use CLR integration features.

Additionally, you may see the above error messages with SQL Server 2008 features which use SQL CLR internally, like DMF (Declarative Management Framework) and spatial data types.