I would start with the description itself.
"This spell creates an invisible, mindless, shapeless force that performs simple tasks at your command until the spell ends. The servant springs into existence in an unoccupied space on the ground within range. It has AC 10, 1 hit point, and a Strength of 2, and it can’t attack. If it drops to 0 hit points, the spell ends."
The first thing I noticed is that the Unseen Servant spell doesn't summon a
creature, it creates a
force. Therefore, it doesn't have Actions, it doesn't roll Initiative, it doesn't have a stat block, it doesn't get save throws, and it can't be targeted by things that can only target creatures. It isn't alive, it doesn't eat or sleep or breathe, it doesn't think, it doesn't have a metabolism, etc., etc. This is all important, so I wanted to say that first. A lot of people want this to be a pet or a creature, and it's not. It's basically an object with a movement speed and Strength score.
So the first question that came up was the OP's question: can it fly? I'd say no, an unseen servant cannot fly...at least, not per the rules. The servant can
move but it doesn't have a Fly speed. So just like everything else in the game that can move: no fly speed, no flight. Pretty straightforward. (shrug)
Can you cast Fly on it, to make it fly? No, because the Fly spell only targets creatures, and this isn't a creature.
Then there were follow-up questions about the Unseen Servant setting off pressure plates and activating traps. The Servant isn't
incorporeal, it is only
invisible, and those two things are not the same. So I would rule that it can indeed activate a tripwire, especially if you hand it a broom and command it to sweep the floor ahead of you as you walk.
While an Unseen Servant doesn't have a described weight, it isn't described as "weightless" either. This is true for most things in the game: wolves and flying carpets, for example, don't have a listed weight but that doesn't mean we can assume they are weightless. So I would rule that an Unseen Servant could set off pressure plates...especially if you command it to carry a bucket of water or whatever. If the precise amount of weight needed to trigger something is ever important (it usually isn't), something strange is going on and I'd just go with whatever I think would be more interesting at the time. Sometimes its fun to have the trap detonate at a distance, hinting to the players that the corridors are very dangerous up ahead...other times it's fun to have it detonate beneath a player's character and watch everyone snap to attention at the table.
What about traps that have proximity triggers and alarms and stuff? Well, if they are keyed to detonate when a
creature approaches within a certain distance, they won't trigger. Because remember: an Unseen Servant isn't a creature. It's not going to be much help if there's a medusa waiting in ambush up around the corner.