Totally unacceptable. My ex-husband is living with a woman who is like this now and I can't talk sense into him. She goes through his texts, emails, FB and has even pulled the numbers of all his contacts out of his phone and started calling them. This is the mildest of the bizarre behaviors she's since displayed. That was just how it started.
Later, she went on to snap on him, screaming and yelling in front of our son while he was visiting, now our son is not allowed over to his dad's house because I don't want him exposed to that. She went so far as to threaten to hurt herself and then call the cops saying he did it. He's throwing away a 13 year relationship with his son over a woman who acts like a psycho - The first signs? She thought she was justified going through his personal stuff. He never did anything remotely like cheating - she is just that insecure... pitiful and potentially dangerous!
In my opinion a snooper is a huge, gigantic RED FLAG that is pointing to future abuse. If someone has to invade your privacy to feel secure and/or to control you that is not a healthy relationship.