The shooting is being investigated by the FBI and the Minnesota Bureau of Criminal Apprehension, working together.
If ICE were being run honestly and competently, ICE would also conduct an internal investigation, but few expect that to happen.
A court case could be federal or state. Few expect a federal case, because Attorney General Pam Bondi would probably interfere with any attempt to bring a federal case; even if such a case were successful, President Trump has already decided the outcome in his own mind and would probably pardon anyone convicted of a federal crime. The state of Minnesota could bring a case, but it's hard for a state to prosecute a criminal case against federal agents; it has been done, but such cases are rarely successful.