I only know of one set of rules for Cutthroat so that's what I'm going to post.

The Scratch

    Establishing a scratch in Cutthroat. A scratch is determined when:
        · the cue-ball is pocketed
        · or is knocked of the table.

The Break

    In cutthroat the break is done form anywhere inside the kitchen any break counts, if you break and scratch the game continues the turn is passed on and the other player has ball in hand in the kitchen. If you break and nothing goes in the turn is passed on. If you don't hit the rack or you completely miss and scratch you have two more chances if you still don't hit it the break goes to the next person. The balls are divided into three groups and the game requires three players the groups are 1-5, 6-10, and 11-15. If a person sinks a ball on the break that is his or her group if he or she sinks more than one and they are in two different groups or three he or she has the choice in between those groups. After that the object of the game is to pocket your opponents balls and and have at least one of yours still on the table.

The Shot

    After that is established the shot is next, (all shots must be called) the breaker continues to shoot until he or she misses or scratches if he or she does scratch one ball from each of the other players groups that has been pocketed will be taken out and placed back on the table at the footspot or behind that. If you accidentally bump the ball or move it it counts as a shot and the turn passes on. If you scratch the other player gets ball in hand in the kitchen. If a player pockets his or her own ball that player loses his or her turn. If a player has no more balls from his or her group left on the table that player is out of the game until someone scratches and one of their balls is placed back on the table.

EndGame

    To win or "end" the game you must have pocketed all of your opponents balls and have at least one of yours left on the table.