The most common fin systems I know of are the klick fin and the US finbox system. Independent of the system, will most SUPs be equipped with standard touring shaped fins.
The touring shape gives solid directional stability and also increases the stability of your board. Nevertheless, a bigger fin will lower your speed.
A flatwater/ race fin will increase your tempo and allows quick maneuvering and turning. It also comes in really handy when the water level is very low without making your board wobble.
However, my favorite fin, especially in summer, is the grass fin. The shape of the fin makes it hard for sea weed and water plants to attach. The grass fin has still excellent directional stability, supports turns a bit better than a touring fin and can also be used in shallow water.