Pride, thy name is Sherlock Holmes. The man is definitely full of himself, and he's actually justified for the most part. We can see why Lestrade gets so aggravated with Holmes at times. No one likes someone who is never wrong, and Holmes's never-ending winning streak has makes him downright insufferable at times. Holmes screws up on occasion, of course. He sometimes is slow on the uptake, and people get injured or even killed. But he never fails to figure things out in the end. Holmes is super confident and is unapologetically arrogant. But on the whole, pride isn't necessarily a good thing in these stories. The old maxim of pride going before a fall holds true for characters like the cocky Lestrade in the "Norwood Builder" case, and the overconfident Milverton, who remained full of it right up to the instant he gets shot. Pride may be a risky trait to have in mere mortals, but not so much for Sherlock Holmes. Which begs the question as to whether or not Holmes is really all that human.
Lestrade and Holmes occasionally butt heads not just because Holmes aggravates Lestrade, but also because Lestrade has a lot of pride, just like Holmes.
Though Holmes is the obvious candidate for "most prideful resident of Baker Street," Watson isn't too far behind him, if his reactions to criticism from Holmes are anything to go by.