Unsure whether this is unpopular or not, but I think every Legendary/Mythical should have a signature move - it just makes sense. So obviously I liked that the Lake Guardians and Phione gained Signature moves in L:A, that were kept in S/V.
There are a few exceptions where it canonically makes sense not to have a signature move, which I’d be ok with. Namely Mew (learns all moves), Type: Null (unevolved and power restricted), Cosmog and Cosmoem (unevolved and unpowered), Meltan (unevolved, evolves by fusing) and Kubfu (unevolved, hasn’t trained yet).
In Gen II Ancientpower was learned (by level) only by fossil Pokemon, Corsola, Ho-oh, Lugia, Mew, and Celebi.
Obviously Ho-oh and Lugia had their own signature moves, but I quite liked this thematically - it makes sense that only "ancient" Pokemon would be the ones to learn such a rare move, and emphasised the rarity of all those Pokemon. It matters less at this point, admittedly, since we're at a point where they just cut a bunch of moves out per game, but it'd be cool to see similar moves restricted to select groupings of Pokemon who, while not directly connected, have a broad association.
Mew of course did later get a signature move in Gen VII in Genesis Supernova, which I'm actually surprised wasn't "translated" into a regular move like Kommo-o's Clangorous Soulblaze was.
These are the current Legends and Mythicals without a signature move.
Legendary Birds (Kantonian forms)
Mew
Legendary Beasts
Side note, I'd forgotten how crappy Aurora Beam was. It was added to Suicune's moveset in Crystal and I'd always thought of it as a quite powerful move because only a couple of other species get it (Remoraid, Shellder, Seel, and Vaporeon learn it naturally while a few other species get it via breeding). I know it's a flavour move for Suicune but jeez it sucks compared to Ice Beam.
...which is probably why Ice Beam is so hard to get in Gen II, come to think of it.
Celebi
Legendary Giants (Regis - apparently thats the official term)
Swords of Justice* (Excluding Keldeo, which has its own move. They formerly had Sacred Sword which was later given wide distribution - they should probably just share Keldeo’s move at this point imo)
Yeah, considering the other three literally teach Keldeo Secret Sword you'd think they'd know how to use it too...
Slight tangent - it bothers me that Blacephalon is the only UB with a Signature Move (do all or none). Similarly with Iron Treads amongst the non-legendary Paradoxes (ofc the Beast and Sword Paradoxes should have them), either do all or none. Or hell do just Roaring Moon and Iron Valiant - counterparts seen as above the others.
There are a few examples of Pokemon getting a signature move but their counterpart(s) not having one - Galarian Weezing has a signature move but Kantonian Weezing doesn't. Hitmonlee had a bunch of kicking moves that were all its own for a while, but Hitmonchan never had an equivalent punching move - even Mach Punch was shared with other species from the word go. As noted, Regieleki and Regidrago are the only ones of their group to get a signature move; Tepig was the only Unova starter to initially get one, and some psuedo-legendaries got one (Dragonite, Metagross, Kommo-o, Dragapult, Baxcalibur) while others didn't (Tyranitar, Salamence, Hydreigon, Kalosian Goodra).
Age-old question really, but in these instances I wonder whether it's a case of "we couldn't really think of one" or deliberate nerfing - Stakataka for instance didn't really need a signature move IMO, but then on the other hand Tyranitar could definitely have used one in GSC. Hard to say.
It's definitely the case that they just don't seem to be bothered going back and giving older Pokemon their own moves - pretty much none of the original forms of regional Pokemon like Braviary or Stunfisk get their own move to match their counterparts. Unless you already had a signature move (like Exeggutor and Marowak) you're shit out of luck.
Scanning Bulbapedia's article on signature moves... Kinesis has been the Abra line's signature move forever, and hey the Machop line also had a signature move - Vital Throw, which... mm, yeah, they were the only ones to learn in Gen II. They surrendered that pretty much immediately though (I'd have thought Cross Chop was more associated with that line but turns out Mankey had that from the off as well! Interesting).