One situation I've noticed happen to me sometimes is that in solving a few problems I find that there's a step I find myself repeating, and which I'd rather have as a lemma stated/proven once and cited thereafter.
A fairly recent example of this happening to me is in differential topology. One statement which I've packaged myself and use every now and then is the following:
Let $M$ and $N$ be oriented $n$-manifolds with $M$ connected, and let $f:M\to N$ be a local diffeomorphism. If $Df_x$ is orientation-preserving for some $x\in M$, then $Df_y$ is orientation-preserving for all $y\in M$.
I thought of it because I realized the proof for a problem on my pset about $\mathbb{RP}^{2n}$ never being orientable basically boiled down to that, and later found myself using it in my (admittedly unnecessarily complicated) proof that compact $n$-manifolds do not have an immersion into $\mathbb{R}^n$. This particular instance isn't terribly inconvenient since it's quick to justify and the statement suggests how the argument will likely go once you put it like that. However, I am curious about whether/when a fact isn't really well-known enough to just cite it straight out and still seems to creep up a decent bit.