To answer this question, I should note that Shakespeare was born in the 1564, and died in 1616, which means he lived in the late 16th century and early 17th. This was a large time for both Discovery and Colonialism, and so firearms rapidly became a major form of both trade as well as usage by colonialist. Muskets were definitely the most common type of gun back then, as well as forms of hand cannons.

Match locks and wheel locks were heavily common at this time, but the fact that many of the firearms were probably owned by armies of a state rather then the citizens can create doubt that Shakespeare ever owned a gun, especially being a playwright himself.

If in fact he did own a gun, which is not likely at this time period because gun ownership rights prinarily became a problem in the 1600s and hunting or even defense consisted of swords or spears until the rise of firearms later on, he probably owned a early form of a musket such as a wheel lock and so on.