I ride rail trails almost exclusively.

I agree with Bev that if it is paved then a road bike is fine.

If it is mainly smooth dirt, and some that started out with a fine gravel have been worn down to a smooth dirt, then a road bike is still quite usable. Although having a 28mm-32mm tire does help smooth the ride a bit. A rigid fork hybrid also works well.

If it is fine gravel, then I find a 32mm tire on a bike (road or hybrid) that has a carbon fork is still okay. I've seen some still using thin tire road bikes on these surfaces, so that is still doable, but I don't find it comfortable. For example, I've ridden a couple of flat bar hybrids that had 32mm tires, carbon forks and carbon seat stays on graveled paths and it was an acceptable ride.

The popular choice on graveled paths is a hybrid or comfort bike with suspension. You don't need a heavy duty suspension as there is no need for a fork that can absorb a major impact. My standard rail trail bike is a Trek 7600 hybrid with suspension fork, suspension seat post and 38mm tires. The ride is very smooth and comfortable. With the suspension, you could get by on thinner tires, like down to 28mm.

A lot of people do use mountain bikes too and they are fine. You really don't need tires that wide or suspensions that heavy duty or gearing that low, but they ride fine.