Are there standardized replacements for these little hex set screw studs used to hold towel bars, toilet paper holders, etc together?

The one for the chrome towel bar below fits a 1/16" Allen wrench, and the one for the bronzed toilet paper holder fits a 5/64" Allen wrench.

The problem is something in there is stripped, so they start free-spinning without tightening any more after a certain point. But, where can I just find simple replacements to try subbing in?

 

About any hardware store will have small drawers in the hardware aisle (nuts, bolts, screws, etc) containing specialized items... and they will have various sizes of both metric and standard set screws. Take yours with you.

 

If its the female threaded stud inside the towel holder that's stripped, you'll need to just buy a new towel holder. Its doubtful that the towel holder will come apart, or that you could get a replacement stud, to replace the female threaded stud inside.

 

If something is stripped it's usually the fixture. It's usually just cheap zinc (pot metal) and it corrodes and strips easily. You can remove the screw completely and look at the threads. I'm betting that after you wipe them off they look good indicating that the fixture is stripped.

If you want to buy new screws they are called "set screw". As XSleeper mentioned they are available in most home centers and hardware stores. The gotcha will be finding the correct size as many ANSI Allen wrenches will fit in a metric screw and vice versa. So, if you buy some set screws and they just don't feel right when you try to screw it in STOP. Think about it and feel carefully to figure out if you just cross threaded the right screw or if need to get the other type.

 

Even if you find that the fixture is stripped you could likely buy the next size larger set screw along with a tap, and chase new slightly larger threads in the fixture to match a new slightly larger set screw.