Creatures and Magic Items have auras, which spells like detect evil and detect magic rely on to function. Do spells have auras, and, if so, are those auras distinct from the aura of the target(s) of a spell, when such targets exist?
All spells have auras, and these auras are distinct from the auras of targets.
There are a number of sources that support this:
1) detect magic consistently refers to auras as stemming from two fundamental sources: spell-like things (not spell-like-ability-like things, just spell-like things) and item-like things. This shows that spells have auras.
2) detect evil specifies that the number of evil auras is the sum of the auras of evil creatures, evil objects, and evil spells, which shows even more clearly that spells have separate auras
3) In 3.5, detect psionics states that "A psionic aura is given off by any active or permanent power, or during the use of any psionic feat." Since the auras are given off by powers but during feats, it's pretty clear that the former possess distinct auras from the characters employing them. While psionics is not magic, and powers are not spells, the mechanics for dealing with them are essentially identical, and this provides strong indirect support for the idea that spells have distinct auras. First-party-psionics in Pathfinder is completely different than 3.5, so this doesn't really apply there.
4) Clerics have a class feature in both editions that makes their aura super powerful in certain alignment directions. If spells are tied to their targets, any spell targeting an aligned Cleric would gain an alignment descriptor based on the targeted Cleric. This would thusly be an extremely consequential interpretation, as it means Clerics cannot target the Clerics of those diametrically opposed to their faith directly with any of their spells.
5) In Pathfinder, the Mask Dweomer line of spells specify that spells definitely have their own auras, since you choose a spell affecting the target and alter that spell's aura (not the target's). Furthermore, that Mask Dweomer hides itself from magical detection implies that spells hiding other spells probably need to hide themselves to be hidden from things that search for magic auras, though it doesn't explicitly say this and Pathfinder has a history of having options grant you the ability to do something all characters can already do by default and lesser versions of said default options, as well as specific immunities to consequences of rules that don't actually have those consequences, so that in particular isn't as compelling as it could be.