For use with high burst damage (which generally means low Fire Rate and high damage per shot); Weapons you're aiming to one-shot enemies with.
Swap elemental mods to suit your enemies. Radiation and Corrosive are generally the preferable types due to their innate armor penetration, maximizing your one-shotting potential; Corrosive for general use and Radiation for Alloy Armor.

Vile Acceleration and Primed Cryo Rounds are flex-slots.
Options for flex-slots:
Riven is an obvious choice if you have one.
Faction(Bane) - High damage increase, but only vs a single Faction. Mainly useful for long endless missions; not recommended for general play.
PCryo Rounds - High damage increase vs low or no Armor.
Vile Acceleration - Medium DPS increase, with the downside of making your ammo economy worse. Performs better the higher the Shooting:Reload ratio of the weapon is.

Good Exilus mods

Vigilante Supplies - The only option that provides a tiny increase to your damage, while also bringing you extra ammo.
Stabilizer - Nice for reducing Recoil on weapons where that is an issue.
Twitch / Tactical Reload - Nice utility.
Terminal Velocity - Good for weapons with damage-falloff, but useless on all others.

Riven

Riven stat priority: CD = CC > FR = Dmg > MS = Tox/Elec/Heat
CC > CD if below ~90% final CC cuz HunterMunitions, else CD > CC.
Riven replaces mod with same stats, in reverse priority; i.e: replace the mod with the lowest priority stat your Riven has. But don't replace Galvanized Chamber even if Riven has MS.

Arcanes

Merciless is the easy all-round option that is especially effective on low-ammo and weapons with low shooting:reload ratios.
Deadhead is the more niche option for precision weapons where recoil might be an issue.
Dexterity is mainly useful as a stat-stick for your melee by increasing Combo Duration without taking up a mod slot in your Melee weapon.