Pushes do not tend to score points in combat sports points based combat sports. In Full Contact combat sports they gain little advantage unless you are able to either take the opponent to the floor or gain a position advantage from them.
However in Chinese martial styles and their derivatives (which are not 'combat sports' but are fighting methods) pushes are fairly common.
In one of my first Shorinji Kempo classes we were shown how to adapt a block and punch technique called uchi uke zuki into a block and push technique.
We also talk about Tai Atari where you essentially body slam an opponent.
(Tai means body Atari is a word from the board game go similar to the word check in chess)
In either case the trick to these techniques is to use the weak points in the opponent's balance. There are 6 of these in a normal fighters stance:
Draw a line from the smallest toe on the left foot to the heel on the right foot. The ends of this line extended out are two of the weak points.
Similarly Draw a line from the smallest toe on the right foot to the heel of the left foot. the ends of this line are two more weak points.
Where these lines meet is the center of balance. At 90 degrees to a line passing through this point and each foot is a new line. The last two week points are on this line. (often called the 'tip of the triangle' by martial artists of all disciplines)
If you push someone along one of these lines they will stagger. Push them fast enough they may even fall over. This is very useful in self defense as it will give you a moment to run.
I believe some forms of aikido also teach similar pushes, but I am less familiar with this style.