Shotguns have been around for hundred and hundreds of years, easily predating the 1500's. Most of the modern shotgun gauges began to appear between 1880 and 1920.
The rounds available circa 1900 include, the 4 (4 bore), 8, 10, 12, 14, 16, 20, 24, 28, 32, and 410 gauges. Currently, the 10, 12, 14, 16, 20, 28, and 410 gauges are still made by most ammunition manufacturers. The 12, 20, and 410 are the "popular" ones, but by far more 12 gauge ammo is made then all the others combined.
The 4 gauge, also called the 4 bore, and 8 gauge were discontinued in 1918 after a law was passed banning all shotguns larger then 10 gauge in size. These were huge diameter barrels and launched huge loads. The 4 gauge was used in Africa, frequently. It could launch a 1750 gr slug...1/4th of a pound !!!, to 1880 gr slug at 1300 to 1400 fps. Even the smaller 5 gauge could launch a 1250 gr bullet at 1500 fps....that's 6300 ft-lbs of energy.
The 16 and 28 gauge are still being made, but are for specialty purposed. The 28 gauge is popular for skeet shooting and the 16 gauge fit a niche for some people, as does the 10 gauge. Modern 12 gauge rounds almost equal 10 gauge loads.
The 14, 24,and 32 gauges were all obsolete long before the start of World War 2. There were made obsolete by 12 gauge, 20 gauge, and 12 gauges.
The most popular shot gun round is the 12 gauge. There are short shells, 2 3/4, 3, and 3 1/2 inch rounds available. These can match or equal 410, 20, and 10 gauge loads.