Technically, there weren't corsets known as sports corsets until the 20th century - in the 19th, it was more common for advertisements and magazines to talk about summer corsets (made of a lightweight but non-stretchy material, often some kind of mesh, or just strips of twill tape), bicycling corsets, riding corsets, etc. For example, in this editorial piece from Table Talk in 1896:

Perhaps the most ingenius of these special-purpose corsets is the one made for the athletic girl. It is a girdle-shaped, elastic affair, hooking in front under loops that lace for tightening or loosening purposes, and is ideally comfortable for out door sports. For the horsewoman the only corset recommended is short over the hips, ending in a wide elastic band. For the saddle this style is pronounced exceptional. The bicycle corset is ventilated, made of light weight coutille, with side steels and elastic bands, and the corset for the decollete bodice is short and low rounded over the bust.

They used some elastication in corsets at the time as needed for the specific activity (some riding corsets were just very short ribbon corsets that ended high on the hips and below the bust), but the actual term "sports corset" or "athletic corset", implying that it's made to be appropriate for all sports use and all types of bending and movement, doesn't really come about until around 1900. Ferris's Athletic Corset Waist is the first I'm aware of. Ferris had already been making a plain corset waist for women and children - a corset waist being a kind of softer replacement for a corset, generally buttoning up the front, with buttons on the waist/hips for attaching petticoats - so it's not too surprising that they'd be the first to get into the game of producing a general-purpose athletic corset (waist) as well. The main change from Ferris's usual corset waist (which Spirella, another manufacturer, soon picked up) was elastic or lacing all the way down the sides. Accommodating deeper breaths/more extreme movement and washability were the primary factors in making an athletic corset waist.

While there are exceptions, in the 1910s and early 1920s the corset waist concept was discarded, and the sports (now sports/dancing) corset became more corset-like again, with a slightly shorter busk, elastic panels over the thighs/at the sides, and/or elastic panels at the top in front to stretch when the wearer bent. (Pictures, pictures.) During the 1920s, as elastic became a bigger part of corsets/girdles, you see more corsets advertised as being good for sports, rather than being specifically for sports.

So, to go back to your questions, the sports corset wasn't just ventilated - that was a more common feature of the summer corset in general, which would be more popular where it was warmer - but usually had a certain amount of elastic for stretching. As to how common they were ... outside of the athletic corset waist era, they can be hard to spot. Is this a sports corset? It has elastic panels down the front. This one has a short busk with buttons over it, like in one of the ads above. This one has elastic panels over the thighs. Because elastication was becoming more and more commonly a part of corsetry, these could have been intended specifically for sports or could have just been corsets that you could dance and ride in. Then there's also the fact that the sports the corsets were advertised for were generally associated with affluence and leisure - horseback riding, tennis, golf. These were just not on the table for the majority of women. So I don't think they can be described as common, but probably any woman who was able to do those things had a corset that was intended to worn while doing them.