Missiles, and special forces will not target oil rigs. Also, AI vessels will not attack oil rigs. Also, for some reason, you can't send a targeting solution on them to the weapon configuration screen too.
Launching a carefully manually aimed TLAM at an oil rig results with the missile passing through the rig like a ghost. The Special Forces just crash into the oil rig, take damage from the collision and just sit there.
I'm pretty sure this is some kind of bug and I've posted about it on the Sonolysts technical support board, but it seems that they don't look at those boards very much. I think that question has been on the board for about a month with no reply. I'm not sure if you can correct any of this in the new LWAMI, but if anyone can track down the issue if it's fixable with the database, you guys can.
Also, as I said before, someone mentioned that oil rigs should generate some kind of detectable sound with all the pumps and machinery they have running. It would be interesting to find out if they do.
Also, a fun way to add an underwater lab/research facility is to simply go into the database and change the oil rig's position setup from "Float" to "Rock" It sets the deck of the rig on the floor of the ocean where it looks pretty convincingly like an underwater research facility looking thing.