NRA Highpower is a good place to start. The most common matches you'll find are generally filled with Match rifle (any centerfire, iron sights with the ability to do a reload) or Service Rifle. They are usually shot offhand, sitting and prone. Long Range (600yds prone) is prone only, again, iron sights. There are also F-class events (any rifle, any sight, bipods allowed, etc).
Silouette is offhand only. I'm pretty sure they use scoped rifles.
Many ranges have Tactical rifle events which will vary from one place to the next.