As others have mentioned, there is so much variation, the best thing really is to Google the specific requirements for your place of interest. Or at least Google the contact information at the city hall in the town you want to get married in.
My experience: I lived in Massachusetts, and two towns in the same county had different ID requirements. Literally, I could drive from one town to the other in about 10-15 minutes, they were both in the same county, and the requirements once I crossed the town line would be different and one set of documents that would be satisfactory in one town wouldn't suffice in the other. It wasn't even a regional variation, it was literally between neighboring towns! I could probably even walk from one to the other if hard pressed to.
Once you know where you want to be married, contact the clerk and inquire as to the specific requirements (documents, waiting period, would your UK spouse-to-be need any special documents, etc).
The variation I talked about earlier - in one town you could use one document to prove both your age and identity (ex. passport). In the other town, you needed two separate documents to independently prove your age and identity (ex. passport for ID and birth certificate for age). It's not a huge difference, but if DH had come over from the UK with only his passport and we had arranged to get married in the town that required two documents, we could have been in big trouble. So it's worth it to check specifically in the town/city hall you're planning on getting married in to get the info on the specifics.