No, I wouldn't because I have been the one snooped on by a partner and I HATED it. I think if you've to the stage where there is so little trust that you feel the need to do this, then you need to seriously look at your relationship and not their phone. Beside which, if you find nothing will that settle the suspicions? Unlikely if you have so little trust and cannot talk to your partner about your worries.
My ex-partner go through my phone, tried to get into email account (including work account which has confidential emails in) because he was convinced I was having an affair. I wasn't. Obviously, he found nothing as there was nothing to find. But it didn't resolve his suspicions so he kept on searching. I found out as I caught him trying to access emails. I was furious and hurt. I also felt my privacy was completely and utterly violated. I ended the relationship as not only was it clear he didn't trust me, I no longer trusted him. And without trust then you have no foundation for a relationship- so what's the point? I just couldn't forgive him for betraying my trust and breeching my privacy. So you also run the risk of damaging (or ending) your relationship if your suspicions prove to be unfounded.
A friend's partner checked their phone and found out a message sent to me after they'd had a fight (along the lines of "x has really upset me- can I come round? He's such a prat") and he went ballistic because a) she had the audacity to talk to me about him and b) he didn't like reading negative things about himself. I think, if you look at things you aren't meant to then you can't complain if you read something you don't like.
I honestly think if you have got to the point that you are so suspicious of your partner that you have to snoop on them, your relationship is in serious trouble anyway- either because they are up to no good or because there is no trust between you. If you have suspicions then speak to your partner. If you don't think there is any point as you think they will simply deny it (i.e. you do not believe they will tell you the truth), then you should end the relatonship IMHO.