An oil rig is a perfect place for some kind of monster - real or imagined - because the quite limited options to leave.

But research might be a little tricky. I mean, visiting an oil rig is not that easy.

The Devil's Tattoo (2003) (Ghost Rig in England, 2001) is about a haunted oil rig. I couldn't find anything that references this, but I believe it was based on a true (?) story about a rig off Scotland. Generally, horror is my least favorite genre, but for some reason I remember hearing about this film. Per IMDB, “environmental activists encounter a mysterious and deadly presence.”

There was also a Tom Clancy Rainbow Six game that had a rescue mission on an oil rig.

The equipment used to drill “shot holes” for nuclear tests are exactly the same drill rigs used to drill for oil – they just drill bigger holes. On dry land, one of the largest holes drilled for a nuclear test was over ten feet across. A worker actually died when he fell down the 4000 foot hole.

The French do their nuclear tests near Fiji and they use floating derricks to drill their shot holes. Just like on land, you have to stabilize the shaft where it enters the earth and it’s done by a person – a diver for the French. The diver gets sucked down a hole when they cause a previous test to colapse.

So, the ghost of the diver who died while drilling the hole haunts the rig and forces the crew to abandon ship. The authorities believe an eco-terror group is responsible and they send in a special ops team to take the rig back.

How quickly we forget this Eddie Griffin / Vinnie Jones vehicle:

Plot Summary for
Blast (2004) from IMDb:

A terrorist, Michael Kittredge, posing as an environmentalist protester leads a team of highly-skilled mercenaries to take control of an oil rig off the coast of California, intending to detonate an electromagnetic bomb over the United States, striking a sort of "new Pearl Harbor" attack on behalf of enemies to the nation. What Kittredge didn't count on is a tugboat captain, Lamont Dixon, who survives an attack on his ship, and is soon recruited by an FBI agent to infiltrate the oil rig and procure information about their plans... and if possible, stop them. In the process, Dixon meets an eager computer expert aboard the oil rig who helps Dixon even as he gets on his nerves and Lamont suspects he can't really trust him.

What's the movie where the hero escapes from a top secret underground prison only to learn that the only way into or out of it is via an old oil rig?

The oil rig itself may be just the jumping off point as the drillers tap into something valuable. The greedy industrialist sends down a team to retrieve the thing that, it turns out, is one of those Things That Man Wasn't Meant to Know.

i'm having a vision of an oil rig, miles out in the ocean. A huge storm.

And a dead body. And another. And another.

The 'monster' in this case being a very human one.