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The description of the detect thoughts spell says that when attempting to probe deeper into a target's mind (which I assume would be required to detect their class):

the target knows that you are probing into its mind

I want to know to whether there is any way at all to determine an NPC's class without them knowing in 5e, e.g. by adding other effects from spells or magic items, or by a different tactic entirely.

I would be OK with answers that have limits such as detecting some classes but not all.

There are no rules that would explicitly provide a way to do this, because D&D doesn’t want to make a stand on whether or not classes are things that characters are aware of and know themselves. Some things—paladin oaths, warlock patrons, etc.—are pretty explicit and obvious, but others, particularly fighter and rogue are very generic and can lend themselves to a wide variety of characters, occupations, and fighting styles. There is no particular reason why they would necessarily see any commonality between them.

Which therefore means that class may not be a thing to know. That may, depending on your campaign, be purely metagame information. It’s entirely plausible, in fact, that even things like oaths and pacts are not as neat and consistent as laid out in the books, but rather more unique to the individuals involved. It is very reasonable for, say, a sorcerer/warlock to say that all of their power comes from the patron, just that it’s a different sort of pact than a pure warlock has. And so on and so forth.

Since D&D doesn’t want to dictate how any given table handles this, they cannot write mechanics that rely on it being “real” in-character information, so no such detection spell could be written.¹ If classes are real, known concepts within your campaign, you will have to ask your DM for details on how you can know and recognize them in character.

1. _{This applies to a lot of things in D&D, and always has, though 5e is arguably trying to go farther with it than other editions. However, they aren’t doing so perfectly—while they often say they support a variety of playstyles, they sometimes seem to forget that and write mechanics that presuppose one particular style. But this is one particular case where they haven’t done that, and personally I suspect they won’t.}

Generally no, but sometimes you could get partial knowledge of this

Only because the OP said it would be open to limited solutions, we could consider using Detect Magic

you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any

So depending on whether your DM rules that a spellcaster counts as a "creature.... that bears magic", you would know that they are a spellcaster, though in 5e this doesn't narrow down class very much, considering things like Arcane Trickster and Eldridge Knight.

Furthermore, almost all wizards have an Arcane Tradition that dedicates them to one school of magic, so it could be argued if they are a wizard, you would pick up their school, and since no other classes have a school, you could infer they are probably a wizard.

(Your DM might however rule that a person does not "bear magic", even when they are a spellcaster, unless they are actively concentrating on a spell. Their magic doesn't "come out" otherwise.)

Finally, via the same spell, you might detect a magic item in the person's possession, and then if through role-playing you could get enough observation of it to make a successful Arcana check, then you might know if it requires attunement by a certain class, and again could infer (though not with certainty) something about that person's class.

Battle Master Fighters know (some of) this information

The Battle Master Fighters gain the following feature at level 7 (PHB, p. 73-74, bold added):

Know Your Enemy

Starting at 7th level, if you spend at least 1 minute observing or
interacting with another creature outside combat, you can learn
certain information about its capabilities compared to your own. The
DM tells you if the creature is your equal, superior, or inferior in
regard to two of the following characteristics of your choice:

• Strength score


• Dexterity score


• Constitution score


• Armor Class


• Current hit points


• Total class levels (if any)


• Fighter class levels (if any)

So you could discover if the NPC had class levels and fighter levels: this isn't foolproof. If they have none, you might be told "fewer" by the DM and not given more information. But if they have as many or more class levels or fighter levels than you, you'll definitely discover that.