The spell says:
The servant springs into existence in an unoccupied space on the ground within range.
Does the servant need to stay on the ground, or can it move through the air?
Ideally I'm looking for rules-as-written, although since the spell definition does not explicitly say, it would be reasonable to perhaps reference other rules than the spell. Interpretative answers, answers referencing previous editions or lore, or from personal experience might be useful.
It's spelled out fairly clearly in the spell description. The description provides a straightforward test to answer all such questions. An Unseen Servant can perform simple tasks that a human servant could do. A human servant cannot levitate, therefore neither can the Unseen Servant.
Moreover, the intent of the spell is quite clear from the tasks described. The tasks are fairly complicated - mending, serving food, and starting a fire all require some modicum of skill. But they all are consistent with what might be demanded of an ordinary manservant. None of the possible tasks suggest any physical abilities (such as levitation) beyond those of an ordinary human.
In sum, the spell creates exactly what it says, an invisible servant.
As was noted in this answer to this question, the Unseen Servant is not a creature. What it does and how it acts is pure magic. The granular detail you are asking for is left for a DM's ruling.
Since you asked for lore and previous editions, let's look at:
AD&D 1e (TSR, 1978, PHB p. 69). It was a first level Conjuration spell as it is now.
The unseen servant is a non-visible valet, a butler to step and fetch, open doors and hold chairs as well as to clean and mend. The spell creates a force which is not strong, but which obeys the command of the magic-user. It can carry only light weight items, a maximum of 200 gold pieces weight suspended, twice that amount moving across a relatively friction-free surface such as a smooth stone or wood floor. The unseen servant cannot fight nor can it be killed, as it is a force rather than a creature. It can only open normal doors, drawers, lids, etc. It can be magically dispelled, or eliminated after taking 6 hit points of magical damage.
From the SRD for D&D 3.5 edition, the description is very close to the 5e description, using the words "invisible, mindless, shapeless force." It is Conjuration magic, cannot attack (has 6 HP), has an effective strength score of 2, and has a speed of 15'.
It cannot be killed, but it dissipates if it takes those 6 points of damage from area attacks. It gets no saves against attacks.
It appears that "it's not a creature" has remained as a governing principle. Unlike a creature, the 3.5e Unseen Servant does not have saving throws.
Could an Unseen Servant (a magical force) move through the air (while staying within 60' of the spell caster?) As the rules do not specify, the DM can rule either way. For a mundane example, have the US close the shutters on a window that is out of reach.
Could the Unseen Servant reach/move up to that window 15' feet off the ground to close the shutters? That's a household function within the bounds of the spell and less complicated than mending clothes. I'd rule that it could, but another DM might not. A valid ruling would be that the US needs to get that pole in the corner and use it to reach up and close the shutters. A regular servant would do likewise.
It's magic.
This spell creates an invisible, mindless, shapeless force that performs simple tasks at your command until the spell ends.
There is no rules-as-written answer to this. Other than the general rules on ranges, savings throws, and magical attacks, every spell in 5th Edition D&D has the entirety of its rules contained within its description.
If it doesn't say the unseen servant can or can't leave the ground in the spell description, then we must refer back to first rule of D&D 5th Edition, and leave it to the DM to make a ruling on whether it can or can't.
