Unfortunately, if you can't trust him, then you can't have a healthy relationship. Cheating, yeah, it's horrible. But there are plenty of reasons that a relationship falls apart, and if there's no trust, that's a huge one. I know it hurts; and it will. And that stinks. But I would think that it hurts being lied to and not being able to trust him. I think there's an amount of doubt in what you're doing when you "snoop" or you feel you have to resort to snooping to get the truth; and in my opinion, that'd be a bigger pain. It's better to get out now instead of letting things progress more and more and you'll continue not to trust him. And then if things get... more serious... how will that work with a lack of trust? I'm not blaming you; I'm saying you have every reason to be mistrustful. But why keep it up?
If your boyfriend saw the error of his ways (genuinely) and promised not to lie, and actually stopped... would you still be able to trust him? Would it take too long to rebuild that trust? or would at least some small part of you would still not quite believe him.
And yes, it may be little things; and some people lie on occasion over "the little things" - but if he's lying and you're catching him on the little things, then what's stopping him from lying about bigger things? What big things has he got to hide if he's a habitual liar?