This is an Analysis on what there is, Not what Could Or Should be, the analysis is conducted of a Sterile Scenario (no UV, no Trinkets, no Sprite Perks, no Pickup Buffs).

Lets Begin by listing. Specific Sets for Bombs

Bombastic Demo Set:
Defences: Normal-Elemental
Damage Bonus: Very High
Freeze Resistance: Maximum
Our Charging time is the Maximum posible (as it takes all the time to do so). Bombastic doesn't spam, Instead: Charge, Moves into position, Deploy, Explode. The key is that Freeze is the status that Could prevent from getting where we need to place the Bomb, hence the Resist.
Is more beneficial to Damaging Bombs than to Status Bombs, Example: Nitronome, Dark Briar Barrage, Shocking Salt Bomb over Ash of Agni, Topor Tantum, Stagger Storm.

Volcanic Demo Set:
Defences: Normal-Elemental
Charge Reduction Time Bonus: Very High
Fire Resistance: Maximum
Our Charging time is the Shortest. Volcanic Spams, the periodic nature of Fire damage slows the Charging time of not one but several Bombs for as long as it last, the Fire resist reduce the Duration of such problem.
Status Bombs or situation where We don't need a lot of Upfront Damage are when We get the most of This set, having a low damage per bomb, but most amount of bombs at any given moment.

Mercurial Demo Set:
Defences: Normal-Elemental
Damage Bonus: Medium
Movement Speed: Medium
Shock Resistance: Maximum
Our Charging time is also the Maximum posible. Spamming is not easy, but moving is, since Shock interrupt a lot but specially movement, the resistance makes it a lot more easy to manage and to avoid enemies. Damage Bombs get the most of this set, but status Bombs also benefit from the movement speed.

Kat Hiss Set:
Defences: Normal-Shadow
Damage Bonus: Medium
Charge Reduction Time Bonus: Medium
Freeze Resistance: Maximum
Shock Resistance: Decrease Very High
Curse Resistance: Decrease Medium
Our Charging speed is average, mostly because Freeze is a one time damage instance instead of over-time, barely affecting our charging the cold resistance allow us to get into position. Shock and Curse are a burden of the set.

Starlit Demo Set:
Defences: Normal-Piercing
Charge Reduction Time Bonus: Very High
Damage Bonus: Slime: Medium
Sleep Resistance: Maximum
Stun Resistance: Decrease Maximum
Similar to the Volcanic this Set is mostly fitted to Situation where You need several bombs not so much damage per bomb, there is a caveat the Damage Bonus: Slime: Medium sounds like if the set where expressively designed to fight Jelly King, although the Status Resistance is Super Situational and the Weakness is somewhat harsh.

Mad Bomber Set:
Defences: Normal-Elemental
Damage Bonus: Very High
Charge Reduction Time Bonus: Very High
Fire Resistance: Decrease Very High
Freeze Resistance: Decrease Very High
Poison Resistance: Decrease Very High
Shock Resistance: Decrease Very High
This one is tough both Charging and Damage are increased by a lot, the drawback is the exposure to status effects shock and fire are notorious, but Freeze can get us trapped in a bad position longer than we want and Poison reduce damage and prevent healing.
It's a risk to use the set on higher tiers, since Status Dealing monster are more prevalent, but also where it truly shines.

We Continue by listing. Sets that affect Bombs

Arcane Salamander Suit
Damage Bonus: Slime: Medium
Damage Bonus: Beast: Medium

Deadly Virulisk Set / Volcanic Salamander:
Damage Bonus: Slime: Very High

Deadshot Set:
Damage Bonus: Undead: Very High

Dragon Scale Set:
Damage Bonus: Beast: Very High

Divine Set (Divine Veil being the part that provides the buff):
Damage Bonus: Fiends: Medium

Heavenly Iron Set:
Damage Bonus: Fiends: Medium

Valkyrie Set:
Damage Bonus: Fiends: Very High

Fallen Set:
Damage Decrease: Fiends: Very High

Black Kat Set:
Defences: Normal-Shadow
Damage Bonus: Maximum
Movement Speed: Medium
Fire Resistance: Decrease Very High
Freeze Resistance: Maximum
Poison Resistance: Decrease Very High
Shock Resistance: Decrease Very High
Curse Resistance: Decrease Maximum

Almost all of them provides Bonus to Damage but no benefit to charging time, that puts them is the Bombastic type of play with consideration towards what status are they resistant and since are damage specific to enemies Families a consideration to movement and attack pattern.
The 2 distinct cases are Fallen Set: with a Reduction in damage and Black Kat Set: that has similar benefits as the Mercurial Demo Set:.

Chaos Set:
Defences: Normal-Elemental
Damage Bonus: Very High
Charge Reduction Time Bonus: Very High
Fire Resistance: Decrease Very High
Freeze Resistance: Decrease Very High
Poison Resistance: Decrease Very High
Shock Resistance: Decrease Very High
Curse Resistance: Decrease Very High
Similar to Mad Bomber Set the play pattern is Compromise Defensive for sheer Damage per Bombs and Bombs per Moment.