Bard's passive is semi random at the beginning, as the chimes spawn nearby for the first 10 or so minutes.
Draven's Spinning Axes are also only semi random, as where they land is influenced by movement commands (move right, the axe will spin and land right)
Fiora's R is semi random, with the first and last (fifth) strike landing on the original person and the others hitting any enemy champion within range (with a seeming priority on unhit enemy champions).
Gangplanks' Cannon barrage had all randomness removed a while back when the damage was changed to a single hit every second over the entire area instead of RNG cannonballs.
Quinn's passive seems to prioritize enemies last hit by her.
Zac's passive has slight RNG in where his blobs land after a skill, but they land very close to him consistently.
Brand's R is random, unless a target is on fire from his passive in which case it tries to hit nearby enemy champions first
Fiddlestick's E prioritizes unhit enemy champions, followed by unhit enemies, followed by seeming RNG. In practice this means if any minions/monsters are nearby it his an enemy champion a max of 1-2 times.
Heimerdinger's turrets attack the nearest enemy unit, prioritizing enemies that have attacked the turret or Heimerdinger, with an overruling priority to attack any target Heimerdinger has hit with a basic attack.
Including this hee, Zyra's plants work the same way as Heimer's turrets, except without a guarantee on the prioritizing enemies that have attack them/Zyra.
LeBlanc's passive triggers if it is up and she goes below 30% health. The clone is controlled by LeBlanc, meaning there is no RNG there.
Nami's Ebb and Flow seems to prioritize the nearest valid bounce target.
Yorick's ghouls attack the closest enemy creature, overwhelmingly prioritizing champions. Any champion within sight range will be chased to be attacked by the ghouls.
Statikk Shiv (and by extension Luden's Echo) prioritize the nearest enemies to bounce to and do so.
Turrets attack the closest minion by an order. They attack the closest cannon minion, then melee, then caster. Summoned pets are attacked first (voidlings, Tibbers, Zyra plants), champions are attacked last unless the champion is within range and harmed another champion. If reset, they repeat this order.
As a side note, Shaco's E passive on monsters is pure RNG.