At WDW you can bring your bag on any ride. However, at Universal several rides will require you to place your bag in a locker. At the rides where a locker is required there will be free lockers to use for the length of time that you are on the ride. The only type bag allowed on these rides are three pronged fanny packs. Even very small cross body bags are not allowed. These rides are the Hulk, Mummy, Dueling Dragons, Forbidden Journey, Men In Black, and maybe one or two others. I've even seen them tell someone to put their tiny wristlet in a locker or in their pocket. I was told to put my small Vera Bradley crossbody minihipster in the lockers. If you use a fanny pack on these rides it must be small enough not to interfer with the restraints or it too will have to go in a locker. Not that you would, but if you make it to the boarding area and are caught, you will be considered a nonrider and be told to go back to the locker area and have to go through the line again, but they usually catch you as you are going in. I have seen a few trying to sneak a small crossbody bag in under their jacket though, who were caught at the boarding area and escorted off. I've also been told to put my lanyard inside my shirt or in my pocket as I was loading Forbidden Journey. Now to avoid the lockers, when I go to Universal, I just make a point of wearing a pair of cargo shorts and don't carry anything that won't fit into my pockets.
The water rides like Jurassic Park River Adventure, Dudley Do Right, and Popeye's Bildge Raft rides have lockers to keep your things dry, but since bags are allowed on them, they are not free.