What do you do after you've discovered your spouse is cheating and they don't know you know?

You should talk to them about it. If you are married, your rules are different than if you are just shaking. You have a legal contract, if nothing else. I wouldn't have a screaming match. Just sit down and talk intelligently. If it escalates into a fight, nothing will be resolved. You can get through this. Sometimes the marriage stays in tact, and sometimes not, but, especially if you have children, it is worth the effort to work things out and stay as amicable as possible.