I remember one time a friend of mine was grappling with another student in class, he wore a cup that tended to dig into you if he was applying hip pressure.
The guy tapped because it was hurting his spine and told him "Man, your cup really hurts". My friend just looked at him, walked off the mat and said "What cup?"
I tried a lot of variation of cups and cup holders (in compression shorts, in a jockstrap, jockstrap over compression shorts, etc...) in the end, I found an RDX cup in a standard jockstrap worn over underwear but under some MMA compression shorts. Comfortable as anything and doesn't budge, no matter how much rolling around.
I'm using the regular Shock Doctor cup and compression shorts (not the fancy pants carbon fiber ones) and they're super comfy. I don't have any shame about wearing a cup. In my particular gym, the VAST majority of guys wear cups. I have a friend that's been training in BJJ 4 years longer than I have (I just started a few weeks ago) who calls me a bitch for wearing one. I just tell him that natural selection is at work making sure he doesn't pass on his genes. I didn't even consider not wearing one since I spent a little while learning Judo a couple of years ago. The first time I got caught in randori with an Uchi Mata I ended up squishing my left one on the guy's leg. I wore one from then on.
I've never worn compression shorts, what's their benefit for training vs regular underwear?
They're like half rash guard material shorts and half jock strap. They have a pocket that holds your cup in place so you don't have to adjust it plus it holds it where it's supposed to be to begin with so it's a lot more comfortable.