There is no one system of selecting state Supreme Court justices that is used by a majority of the states. One thing that we can say is that in the majority of the states, governors are allowed to appoint justices with some degree of input from one body or another.
There are two ways in which this is done. In 15 states, governors get to appoint justices but must have their appointments approved by some body, usually the state senate. In 16 more states, the “Missouri Plan” is used. In these states, governors select justices. However, they are presented with a list that is drawn up by a nonpartisan commission. The governors must select someone from that list. If they do not, the commission selects a justice.
In most other states, justices are elected. In the majority of states with judicial elections, the elections are non-partisan. Partisan elections are only used in seven states while 15 states use non-partisan elections.