Communication is the key here, as it is with any relationship. Stupid little things tend to be either, as ill suggested, something that will wear on the relationship over time, or a disguise for bigger issues that the two of you are avoiding for one reason or another.
Every argument, no matter how small, deserves to be discussed long enough to find a solution that satisfies you both and prevents it from becoming a future problem. As a story of how out of hand poor communication skills can be, I'm good friends with a couple who fights every night about what to have for dinner. They scream, they cry, they storm off and say terrible things, all because it's a little stupid argument they have every night and so they're both sick and tired of it. However, they both refuse to back down, so it just keeps going. It's very easy to get trapped in something like that without realizing it over the most pointless of things if you doing seek out a solution together.
And hey, trust me, part of moving in together is just rough. Learning to live with another human being is incredibly difficult at the best of times, and there are some stupid little things you're just going to have to let go even if you don't want to. Remember: Nobody wins an argument, and trying is just going to cause more pain.
Edit:
Also, as an afterthought, be sure during your conversations about these things to use terms like "I think" and "I feel" as opposed to "it is" or "I know." You'd be surprised at how much that helps keep feelings from getting hurt.