@Germlat,
Your first question "Is looking through spouse's cell phone spying?" is a yes or no question. The answer being yes.
Your second question "Is it completely inappropriate to look through your spouse's cell phone on occasion?" is a circumstantial and opinionated question. My ex starting showing signs of infidelity: sleeping in the guest room so he could text until the early hours in the morning; staying late at "work"; etc. I knew asking would have gotten me nowhere. So, I snooped. Found out he was fooling around with his coworker.
Your third question "Is it downright obnoxious?" is, again, a circumstantial and opinionated question. For you it may not be, but for your spouse it may be obnoxious. How often are you looking through your spouse's phone? Everyday? Once a week?
Your fourth question "Is it a breech of trust?" is a yes or on question. The answer being yes. Whether it's warning signs from your spouse, insecurities or curiosity that cause the snooping, it's still a breech of trust. You have to ask yourself why you're looking through their phone. Have they given you a reason to believe they're up to no good? If so, address it. Is it your insecurities? If so, address it. If it's curiosity, address it. Tell your spouse your reason and if they hesitate to show you what you're asking for, then get suspicious. We share show much with our significant others that e-mails, cell phone messages, etc. shouldn't be an issue.