All of the severs allow players to be Player Killers, Darktide is exclusively PK, the other servers default to Non Player Killer and also let players do Player Killer Lite.

Live Worlds

Darktide

Frostfell

Harvestgain

Leafcull

Morningthaw

Solclaim

Thistledown

Verdantine

Wintersebb

Beta Servers

Blue

Green

Red

Test Servers

Shadowclaim

This server was available for all players from ??/??/?? to ??/??/?? to stress test some new server hardware. The character database was an exact copy of Solclaim, hence the name Shadowclaim.

Throne of Destiny Open Preview

This server was available for all players from ??/??/?? to ??/??/?? as an open preview of the upcoming expansion pack Throne of Destiny. The character database was an exact copy of Leafcull.

Internal Test Server

As mentioned in numerous posts by developers, Turbine has an internal test server. It is currently not available for players, supposedly. There has been some active players granted access to the test server to test content prior to it being delivered to the general player base.