The most popular are probably rubber reins due to their excellent grip. If you use them, make sure that you don't use saddle soap or any sort of conditioner on them as it can make them grungy. The rubber will eventually wear down over time with use and you will need to either get them re-coated or buy new ones.
Some riders prefer to use web reins. They are not as bulky as rubber reins and are less likely to become slippery when wet than leather reins.
Leather reins, laced, braided or plain, are always an option. They provide a thinner alternative for a curb rein but don't tend to be as grippy as the other choices.
If you are using a running martingale, make sure that your reins have rein stops on them (seen on the reins in this photo). If either of the rings gets stuck on the end of the reins or on the bit, your horse is likely to panic and using rein stops between the martingale and the bit will prevent that from ever happening. Most rubber reins tend to come with leather rein stops already attached, but if your reins don't have them you can buy rubber ones that are quite easy to slide on to the reins. It really is important for your safety not to use a running martingale without them.