An Aasimar Hexblade Warlock (1), Wild Mage Sorcerer (1), Grave Domain Cleric (2), Anything (16) with a ninth level spell slot available can get at least +272 and at most +492 damage on a single melee spell attack

This works as follows:

As pointed out in other answers the Aasimar gets the ability to transform which states:

During it, [...] once on each of your turns, you can deal extra [necrotic or radiant] damage to one target when you deal damage to it with an attack or a spell. The extra [necrotic or radiant] damage equals your level.

Thanks to user @BenjaminHubbar for pointing out that the Wand of the War Mage does not actually add its +3 bonus to damage rolls.
I'm unsure if there is a +3 magical focus that adds that +3 to its damage rolls, but if so, we would surely want that.

The Hexblade Warlock's Hexblade's Curse feature states:

Starting at 1st level... You gain a bonus to damage rolls against the cursed target. The bonus equals your proficiency bonus.

The Grave Cleric's Channel Divinity: Path to the Grave features states:

Starting at 2nd level... As an action, you choose one creature you can see within 30 feet of you, cursing it until the end of your next turn. The next time you or an ally of yours hits the cursed creature with an attack, the creature has vulnerability to all of that attack's damage, and then the curse ends.

Because we are a 2nd level Cleric, we have one use of channel divinity, which we will use on some creature.

Using this fact we could cast the spell inflict wounds which states:

Make a melee spell attack against a creature you can reach. On a hit, the target takes 3d10 necrotic damage.
At Higher Levels. When you cast this spell using a spell slot of 2nd level or higher, the damage increases by 1d10 for each slot level above 1st.

The Wild Magic Sorcerer's Wild Magic Surge table has the following result:

33-34 Maximize the damage of the next damaging spell you cast within the next minute.

We would want to have this Wild Magic benefit available, be in our Aasimar transformation, be using a Wand of the War Mage like above, be using the Hexblade's Curse as above, be using the Grave Cleric's Path to the Grave feature as above, and then cast the inflict wounds spell at ninth level.

Thus we can calculate our new damage:
Assuming we crit we would have 22d10 of damage dice.
The Magic Surge roll would maximize the 22d10 to 220 damage.
The Hexblade's Curse feature would add on 6 damage from our proficiency bonus.
The Aasimar transformation would add 20 damage to our attack.
And then the Path to the Grave feature would double our damage at the end.
In the end we will deal (220+6+20)*2 damage, or 492 damage.

If we did not crit then we would deal (110+6+20)*2 damage, or 272 damage.

This is provided as an additional answer as it only gives it one 1 turn, and is highly situational

Goblin Rogue Assassin 17/Hexblade Warlock 1/Grave Cleric 2: +196 on a single attack

We have a +3 weapon with the heavy property and the Great Weapon Master and Charger feats.

We take a setup round (which is before initiative) using our action on Grave Domain's Path of the Grave on a creature that is larger than us (Medium or larger; for goblin reasons):

As an action, you choose one creature you can see within 30 feet of you, cursing it until the end of your next turn. The next time you or an ally of yours hits the cursed creature with an attack, the creature has vulnerability to all of that attack's damage, and then the curse ends.

Then use our bonus action for the Hexblade curse on the same creature:

As a bonus action, choose one creature you can see within 30 feet of you. The target is cursed for 1 minute. The curse ends early if the target dies, you die, or you are incapacitated. Until the curse ends, you gain the following benefits:

• You gain a bonus to damage rolls against the cursed target. The bonus equals your proficiency bonus. [...]

We now need to enter initiative with the the creature being surprised. On the turn where that creature is surprised we Dash, move 10 feet (or more) towards it, and use our bonus action from Charger to make a melee attack. We have advantage and disadvantage on this attack (but we don't care) and use Great Weapon Master for -5 on attack and +10 damage. We also use Goblin's Fury of the Small:

When you damage a creature with an attack or a spell and the creature's size is larger than yours, you can cause the attack or spell to deal extra damage to the creature. The extra damage equals your level.

When we hit, the Assassin's Death Strike triggers:

When you attack and hit a creature that is surprised, it must make a Constitution saving throw ([...]). On a failed save, double the damage of your attack against the creature.

And we assume the creature fails the saving throw.

Calculation:

• +5 Strength
• +3 Magic Weapon
• +10 Great Weapon Master
• +5 Charger
• +20 Fury of the Small
• +6 proficiency bonus from Hexblade curse
• ×2 from Death Strike
• ×2 from vulnerability

$$(5+3+10+5+20+6)\\times2\\times2=196$$

We can use a Belt of Storm Giant Strength to boost our Strength to 29 giving and additional +16 and/or a Ioun stone of Mastery for an additional +4.

We can also have a Aasimar (Protector or Fallen) and have the transformation active during the attack, however this requires more setup and I wanted to showcase the Goblin feature for this.

Also, yes we have 17 levels in Rogue and are not using the Sneak Attack.

There is no limit to the melee weapon attack static damage

With a Moonblade with arbitrarily many runes, one can get many 93-94 runes which provide (emphasis mine):

When you hit a creature of a specific type (such as dragon, fiend, or undead) with the moonblade, the target takes an extra 1d6 damage of one of these types: acid, cold, fire, lightning, or thunder.

A cleric of the Tempest domain can maximize the damage roll of this absurd number of d6 thunder damages via the Channel Divinity: Destructive Wrath feature:

When you roll lightning or thunder damage, you can use your Channel Divinity to deal maximum damage, instead of rolling.

Since all of the runes cause "extra damage", it counts as one damage roll and deals an arbitrarily large amount of static damage. However, the damage from this is only 6 times the number of runes; as long as you have less than thirteen 93-94 runes, the strategy in the next section can beat it¹

^{1: with thirteen 93-94 runes or more, follow the strategy except you are only a level 6 wizard and a level 6 Tempest Domain cleric, and no Great Weapon Master feat}

The best melee weapon attack static damage²

The best you can do for a melee weapon attack without a very old Moonblade is 126

The Character

The character who can get the most static melee weapon attack damage is a level 20 Protector Aasimar
{ASs: 13, 8, 8, 18, 13, 20} with the following classes:

• Warlock (Hexblade) 1 --------------- {for Hexblade's Curse}
• Cleric (Grave Domain) 2 ------------ {for Channel Divinity: Path to the Grave}
• Wizard (War Magic) 10 ------------- {for Power Surge, ASIs, green flame blade}
• Paladin (Oathbreaker) 7 ------------ {for Aura of Hate and ASI}

...with the following equipment:

• Belt of Storm Giant Strength (attuned)
• Ioun Stone of Intellect (attuned)
• +3 Magic Weapon (any Heavy weapon; attuned)

...whos received specialized Training (as described in the Dungeon Master's Guide) for the Great Weapon Master feat

The Weapon Attack

By using Hexblade's Curse and Path to the Grave on the target of your melee attack (as well as ensuring another target is within 5 feet), you get the following static damage modifiers:

• 3 from Magic Weapon
• 10 from Great Weapon Master (feat)
• 20 from Radiant Soul (Protector Aasimar trait)
• 9 from Strength (Belt of Storm Giant Strength)
• 6 from Hexblade's Curse (Proficiency)
• 5 from Aura of Hate (Charisma)
• 5 from Power Surge (against the primary target)
• all above doubled because of Path of the Grave (Vulnerability)
• 5 from green flame blade (Intelligence against secondary target)
• 5 from Power Surge (against the secondary target)

TOTAL: 126

2: A different build for 196 damage is explained by Someone_Evil in his excellent answer, but relies on a Constitution saving throw, whereas this technique just requires that you hit. Assuming the target fails the Consitution saving throw, his build is a valid substitute for the Moonblade when you have less than twenty-four 93-94 runes.

The best melee spell attack static damage

The best you can do for a melee spell attack is 278 (or 498 on a critical hit).

The strategy for this is explain beautifully in Medix2's answer, so I won't rehash it here.

Zealot Barbarian (20) Total static damage of +34

For this build we use the true definition of static damage; every attack turn and without single target or one a day use restrictions, such as, Hexblade's Curse which is against a sole creature or the Aasimar transformation which is once a day. We use a character with the Belt of Fire Giant Strength (sets strength to 25 or a +7 modifier). This belt is the highest belt that is not legendary. They also have the Great Weapon Master feat. We also have a +3 heavy weapon.

A level 20 Barbarian gets +4 to damage due to rage. This rage can be used 6 times a day.

Path of the Zealot (Xanathar's) gets:

Divine Fury

Starting when you choose this path at 3rd level, you can channel divine fury into your weapon strikes. While you're raging, the first creature you hit on each of your turns with a weapon attack takes extra damage equal to 1d6 + half your barbarian level. The extra damage is necrotic or radiant; you choose the type of damage when you gain this feature.

Great Weapon Master lets us take -5 penalty to the attack roll to get +10 to the damage.

The calculation:

• +7 Strength
• +3 Magic Weapon
• +10 Great Weapon Master
• +10 Zealot's Divine Fury
• +4 Rage

$$7+3+10+10+4=34$$

I believe this is the most reliably useful static damage a PC can get at level 20 and is most in keeping with the spirit of the actual question. This damage can consistently be used in every encounter, on every attack round, against more than one enemy and is not partially from a legendary item, as the OP has stated.