I run cross-country and track and all i ever run in is the regular running shorts with the built in liner. They hold pretty well in my experience as long as you buy the right size. I've only ever had floppin out problems a few times but it wasn't anything serious. Some guys on the team wear compression shorts but i think that they are uncomfortable.
I don't like compression shorts its just too much material and I think it looks terrible when the compression shorts are longer than the running shorts.
I recently got a new pair of running shorts that I just love, the key is to get the right size so the inner liner is not too big, at one point I had bought a bunch of size L but these the liner was totally useless but with the M it fits just right with just the right amount of support.
When I go running on a daily basis I wear briefs or a jock strap.
I got some swimmers jocks since in normal running shorts if your running without a shirt you would be able to see the waistband which isn't a very cool look. Its definitely a hot feeling to have your ass hangout, but with the built in liner, and jock it all feels a bit too restrictive.
The new jocks are made with softer fabric and weaker elastic in the pouches, and straps. They won't last for 5 years but they are a lot more comfortable.
yea i recently got a new bike cup jock that is made of a much softer fabric than my previous one, so I can actually wear a cup if I want to.
I don't know how the manufactures expected cut guys to wear the old fashioned jocks, as the material just always irritated my head too much and i'm only partially cut, I can't imagine being fully exposed in one of those things, I man there probably fine for uncut guys since they don't have sensitive skin exposed to rubbing on the fabric.