This is a very difficult problem to solve. It takes the thieves literally a couple minutes to cut off the cat using a portable saw, so a lot of deterrent methods like cameras, or parking where people might walk or drive by and see them, aren't worth much. The goal is really just to make your car the least attractive target of all the ones near it and hope that they haven't hit your car yet by the time they decide to leave.

When I looked into it, here were the methods I saw. Painting the cat orange and etching your VIN into it discourages dirty scrap yards from buying it, so some thieves will pass yours over in favor of another. There are products like a CatStrap that fasten a very long strip to the length of the exhaust pipe, which contains a loose metal wire so if they hit it with a circular or reciprocating saw the wire moves and they can't get a grip to cut it. Similarly, some people fashion a "cage" of loose wire around the cat--however, you need to make sure the cage is long/integrated enough that it isn't as simple as just cutting the exhaust pipe further down and taking the whole assembly to extract the cat later. Fastening a plate to the car to cover the cat as someone mentioned might work, but you lose the benefit of the loose wire interfering with the movement of the saw so I'm not so optimistic about that. However, every discussion I've seen from informed people and cops about these methods has emphasized that none of these will stop a determined thief with the right tools and experience.

Unfortunately, the best method is more about what you drive and where you drive it. If you apply some of these, also park in a busy area and around a bunch of cars so yours probably doesn't get hit first due to the deterrents. Hybrid cars are targeted either because their cats have more of the valuable materials or because they often have multiple cats. SUVs and other vehicles raised above the ground are targeted above lower cars because they're easier to get under. When mine got stolen, I was super lucky: My mechanic pointed out that my car had just turned 15 years old and no longer has to pass emissions, so they simply welded a pipe in place of the old cat. Can't steal it if it ain't there!

The idea is that if the yard can't unload them fast enough and the police do a surprise inspection, the orange paint (and etching the VIN on the cat is another common strategy) on some cats on the shelves makes it look suspicious, and if they can't explain where it came from, it can get legally rough for the yard. So, ostensibly, some of the dirty scrap yards will refuse to buy cats that have been marked like this. Edit: Or, at least, without more detail and proof about the origin of it.

The Muse just closed, sadly. This past New Year's Eve was their last day. Big loss for Worcester :(