I guess we are talking about in the United States, or the colonies which became the United States and their westward territories?
It is hard to know just how commonly rifles were used as hunting weapons. Statistics are not available. The definition of 'Common" is not precise.
Still, from the available evidence, rifles were fairly "common", particularly on the frontiers, however muskets (generally of the type called "fowlers") were predominantly used.
The fowler was a smoothbore firearm generally lighter than a military musket. They were capable of firing various loads from "birdshot" like a modern shotgun, to bullets. The fowler was often more accurate than a military musket. This was because the bullets were cast to fit the barrel more tightly.
Military muskets were built to fire 50 or more rounds in an engagement, without the black powder fouling the barrel and making further shots impossible until the gun was cleaned. Black powder is very fouling, so this required a bullet somewhat smaller than the barrel so that the gun could still be loaded and fired after numerous rounds. These gaps between bullet and barrel reduced the accuracy of the gun. Military muskets were also generally heavier and more rugged than fowlers, and could be fitted with bayonets.
Fowlers, were designed to be used for hunting. It was likely that only a few shots at a time would be possible against game targets, then the gun could be cleaned. Therefore, bullets were cast with tighter barrel tolerances and fowlers were more accurate than military muskets. Modern firearm enthusiasts who hunt with 18th century fowlers report them accurate to 60 yards against game like deer and bear.
Fowlers were imported in large numbers (The largest numbers being "trade guns" to sell to the Indians for furs) to North America from France, Britain, the Netherlands and Germany, and were also made by American gunsmiths.
Fowlers were most often in .60 or .65 caliber. They were particularly used for hunting deer, and were particularly useful for hunting birds like turkey, pigeons or water fowl like ducks or geese when loaded with shot like a shotgun.
In the Eastern colonies, they were the most common hunting guns (as waterfowl were often the most common prey).
The fowlers were also used for defense, and by the militia, although some militia were equipped with military muskets instead.
Rifles were introduced into the American colonies in the early 1700s from Germany. American gunsmiths, particularly in Pennsylvania, around the town of Lancaster were making their own unique adaptation of the German Jaeger rifle by sometime before 1720.
These "long rifles" (Later called "Kentucky Rifles" - after a song celebrating their contribution to the Battle of New Orleans became popular in 1820) were commonly used along the frontier.
The long rifle was more accurate at longer ranges than a fowler. There is a well documented account of Daniel Boone killing a British officer as he was peeking out from behind a tree at a range of 250 yards (paced out by several witnesses after the battle) in 1778 at the siege of Boonesborough, Kentucky.
The long rifle was generally of smaller caliber than a fowler (.40 to .50) and had a longer barrel and a higher muzzle velocity. These guns were useful for hunting on the frontier, where larger game was more common and waterfowl not hunted as frequently. The frontier was also more dangerous, and the long range rifle was useful in frontier warfare and defense.
The evidence for how common rifles were on the frontier is spotty, but there is enough to indicate that they were fairly common. For one thing, we know of hundreds of gunsmiths who made long rifles in the 18th century, so if many were produced, they must have been used.
Another piece of evidence is that George Washington managed to recruit about 1,400 riflemen from western Virginia and Pennsylvania to join the Continental army. These were all men who came equipped with their own rifles (eventually the Continental army commissioned some rifles to be built, but not until later in the war).
Since we doubt that any more than a small percentage of western riflemen joined the Continental army, there were probably some large multiple of 1,400 riflemen and hunters on the frontier.
If we wildly speculate that 1,400 rifles joining the Continental army indicates circa 14,000 riflemen on the frontier? Then we can say that the rifle was pretty commonly used as a hunting weapon in the west.
There were many more people in the east, however, and most households (especially rural households) had a fire arm, and that was usually a fowler. So, it is safe to say that muskets were still predominantly used.
Sources:
http://www.tngenweb.org/campbell/hist-bogan/KentuckyLongRifle.html
http://www.jaegerkorps.org/NRA/Hunting%20Guns%20in%20Colonial%20America.htm
http://pabook.libraries.psu.edu/palitmap/Rifles.html
http://www.americanlongrifles.com/american-longrifle-kentucky-rifle-story.htm