Are Saturn's rings stable? Saturn's rings contain many moonlets that shape the rings of Saturn. The structures in the rings of Saturn around moonlets are similar to those in protoplanetary disks around newly formed planets, which makes me wonder if the material in the rings will follow a similar fate and form into larger bodies. 

Will the material within the rings of Saturn eventually coalesce into moons, or are the rings of Saturn stable enough to last billions of years?
Most of Saturn's rings are inside it's Roche limit, which means they will never clump together.   Tidal forces prevent this from happening.

small objects that are already together can withstand the tidal forces.    A sufficiently large body inside a planet's Roche limit should break apart by tidal forces.   That may be how the rings formed in the first place, or they may have formed by collision.

Saturn's rings are thought to be relatively young, if 100 million years qualifies as young, but it is for objects inside the solar system.    Being young doesn't tell us how long they will last.   I think the 300 million year estimate may be accurate enough, originally mentioned in the comments, but until that happens, the rings will remain, just grow smaller over time and they'll never coalesce into moons.  

As a footnote, Saturn's thinner outer rings or gossamer rings are outside the Roche limit, and those might, one day, form into moons, or they may be too thin for that to happen.  A certain density is probably required for moon formation.