Amateur Wrestling (Free style and Greco-Roman)
MMA (what you see in the UFC)
Olympic Tae Kwon Do
Sport Brazilian Jiu-jitsu
There are many traditional martial artists that do not like combat sports.
I like it. I like watching matches to study and learn from them.
I personally do not like competing because I don't like hurting people, and I hate injuries.
The more you train and compete in combat sports the higher chance for injury, so my advice to anyone is, while I do think it's important to experience combat sports, once you do, get out of it as fast as you can to avoid injury. Injury is the enemy.
At home alone I train in combat sports. I enjoy how it gets me in better shape.
I go running, sometimes I use the jump rope, I hit the bags, I jump on the tire, I do shadowboxing and so on.
Whether we like it or not, if you take two human beings from anywhere around the world and make them fight, it looks something like what we see in MMA(the sport of cage fighting).
If you give each person an Escrima stick and make them fight, it looks like MMA with sticks.
If you give each person a knife and make them fight, it looks like MMA with knives.
I'm not saying that MMA or combat sports is the single whole and only truth.
I still train in traditional martial arts to this day and will continue to do so until the day I die, but it's important to know that whatever it is that you train in, when you actually apply it in real life in a real violent situation, know that it's going to look something like MMA.
Everybody seems to value the importance of sparring, yet when it comes to sport, suddenly people are against it. Combat sports to me is nothing more than sparring taken to the extreme. So if you think sparring is important I don't understand how you can't see that combat sports is also valid.
People often times criticize combat sports for their rules. The rules are limitations, but fighters are forced to develop specific skills within those limited rules and they become exceptionally skilled. For example Boxers are limited to only punches, however their punching skills are incredible. You get good by limiting yourself, because your focusing on something specific.
If a Boxer fights a grappler, chances are the Boxer will get taken down and beaten on the ground fairly quickly, but that doesn't mean that the Boxer's punches are not effective. They are still dangerous and if the grappler isn't able to take the Boxer down, the match could go the other way.
Then people criticize the Boxer for not being well balanced. A Karate practitioner may say, I will just kick the Boxer, and while that may be true, it doesn't make Boxing obsolete. If you look at the military, not all the weapons are well rounded, many of the weapons are designed to do specific tasks. So you need a good strategy to know when to use what.
Modern military have vehicles and firearms, but soldiers to this day still carry a knife. Even though most soldiers today do not carry swords, if they were attacked in close range by someone with a sword, you bet the sword is still a formidable weapon. If you get into a road rage incident and begin yelling at some guy, and if he has Boxing training, when he throws a punch at you, that is going to be strong punch period.
Even though martial arts can be practiced for many different reasons, strictly combat wise, I divide martial arts into three general categories.
1. Traditional Martial Arts
2. Combat Sports
3. Modern Tactical (includes military styles and urban self-defense)
I highly recommend that you experience all three types so that you get a better more well rounded point of view of combat methods. One isn't better than the other, each one has pros and cons.