This is my take on relic quality for multiplayer games, but almost all of this will apply in singleplayer games too. I am quite new to the game so take my opinions with a grain of salt. Relics are in no particular order within a tier.

S:
Crown of Bureaucracy - 50% extra actions in the early game is extremely powerful and versatile. Even if you do manage to naturally reach the point at which you'd have six orders, this is still providing 30+ orders over the course of a game. Hoping this gets a downside added in the future, as it's a tough argument to ever not take this.

A:
Amulet of Archer's Precision - ranged is the best phase, and prophecy is always useful. Best with Lilith but I can see any leader benefiting from this.
Amulet of Charisma - Charisma is the easiest skill to rush to 4, and has extremely useful boosts. Best with Beelzebub, Lilith or Murmur, as they'd start at 0 otherwise.
Amulet of Soldier's Vigor - Good for a combat focused game. Wrath is probably the weakest stat overall but the extra healing is nice, and high level stratagems are extremely powerful, probably the best way to take Pandemonium early. Best with Astaroth or Erzsebet.
Ring of the Loyal Servant - A great early power boost if you want to start taking POPs and demanding tribute as early as possible.
Amulet of Nobility - Boosts your starting income by 50%, and will help you get further prestige levels faster. Getting a snowballing early advantage is great, though be aware in the long term this is a small boost if you can't leverage it into strength elsewhere.
Amulet of Prophecy - Prophecy is the easiest stat to raise to 4 and I find myself doing so quite often. Best for Astaroth, Andromalius, Erzsebet or Beelzebub.
Ring of the Seer - Good for deceit counterplay. You really don't want your best stuff stolen at an inopportune time.

B:
Crown of Authority - Ranking up faster will get you a solid income boost, but put a target on your back.
Crown of Alacrity - Good solid value, but does nothing to help you win the fights you'll be arriving at faster.
Amulet of Deceit - Deceit doesn't do a ton for you early game, though I guess this allows for spamming resource stealing to slow down the leading opponent. Best on Astaroth, Mammon or Murmur.
Amulet of Sorcerer's Blight - The worst of the three combat boosting amulets, though still quite useful. Best with Murmur or Beelzebub.
Ring of Tyranny - It's difficult to quantify the exact benefit of this, but it's a good boost to your odds of not having to take a 1 value token. Combos well with Mammon and the Amulet of Opulence. A reasonable tertiary relic to take with any amulet.
Ring of the Liar - Only really worthwhile if you want to go all in on spamming deceit rituals. Deceit is extremely good, though, and you should be ready to focus on it any game you're not winning by other means. The only prophecy ritual you really need to resist is Demonic Interference, and avoiding that is not worth a relic slot. Decent combo with Amulet of Deceit.
Amulet of Opulence - A good solid income boost, half the benefit of a prestige rank that stacks nicely with said ranks. Safe and effective choice.
Amulet of Destruction - What's better than spamming duels for prestige? Spamming kill 1 legion vendettas and then blasting them from a million miles away. Probably unwise against Erzsebet's bonus healing. Best with Mammon, Belial or Erzsebet.

C:
Crown of Intolerance - Very fun concept, I will likely try it out, but still has the issue of putting you behind in terms of actual combat potential relative to other relics.
Amulet of Intimidation - I'm sure this can add up to a ton of bonus prestige, but you still need to be in a position where your opponents will feel obligated to accept your demands.
Amulet of Wrath - The early levels of wrath do very little, and it's a tough stat to level to 4. Best with Lilith, Andromalius or Belial. Not recommended.
Ring of Glory - Decent secondary with a combat boosting amulet, but I prefer raw power boosts.
Ring of Spectacle - A good choice for Andromalius, I wouldn't bother as anybody else.
Ring of Tartarology - In the current state where Seek Manuscripts is more likely than not to draw only praetor skills, I wouldn't advise taking this one. Still a decent boost and fun if you like gambling.
Ring of the Arcane - Probably a meta pick if you know an opponent will be spamming destruction, or if you intend to do so yourself. Combos well with Amulet of Destruction or Sorcerer's Blight.

D:
Crown of the Administrator - Basically unplayable in my opinion, go for combat boosting relics to take POPs earlier and get extra prestige that way.
Ring of Command - Investing in a large number of legions is a trap, in my opinion. You'll be spending all your orders just moving guys across the map. Just level wrath.
Ring of Renewal - You're better off avoiding damage in the first place, or leveling wrath for extra healing. Particularly useless with Erzsebet.
Ring of the Collector - Probably the weakest relic in the game. Do not take this.
Ring of Unnatural Prescience - You're not going to be drawing enough events to make this good, and even if you do, just cast them before your turn as regent comes around again. Only real value I can see is to hold onto the really nasty events you don't want to be hit by.

Unranked:
Crown of the Kingmaker - Extremely powerful if you can guess in the first 10 turns who's going to win, and avoid them wiping you out. Better hope they don't realize your strategy.
Crown of the Power Behind the Throne - This one seems basically impossible to win with in a real game. Nobody who knows what they're doing is going to accept a blood vassal offer without first scrying your relics.

3/7:
Increases:
Ring of the seer: B to A
The more I play, the more I think deceit is the best stat. Stealing other people's stuff at the right time will win you games, or lose them if you're the target. Being able to reliably resist rituals from someone at the same deceit level, or higher, is extremely important.
Ring of the liar: C to B
For the opposite reason as the above, this is a good ring, though of course you're not getting as much benefit as an opponent taking seer. Stacking up a lot of strength makes your rituals get past those annoying defensive abilities.
Amulet of Nobility: B to A
Tried this one out a few times, it's the strongest single early-game boost you can get aside from bureaucracy.
Amulet of Prophecy: B to A
This will let you reach level 4 and an extra order extremely fast, so moving it up a tier.

Decreases:
Amulet of Opulence: A to B
The problem with tribute quality is a lot of the time you get mixed tokens that are hard to benefit from. There are usually ways to leverage them effectively, but those ways will often not be what you need or want to do. More tokens beats better tokens, in my opinion.
Amulet of Destruction: A to B
Early game destruction is pretty good, but I don't think it's good enough to not just pay for the levels.