Which items should I buy (or not) in Arena?

I researched so much, I've decided to just post what I've noted as an answer.

The following data were collected according to as many screenshots as I could capture from the $10K Alienware Arena Cup (which took place 19 April 2014, which is important because the developers change items often). Lesser-known teams and names were given less weight. Items are usually enumerated in a meaningful order.

• As with any other game-mode, Tabi are staples for physical characters and Shoes for magical characters.
• Items that build stacks are discouraged.
• Common actives are Heavenly Agility, Greater Sprint, Curses, Greater Aegis, Greater Purification Beads, and Greater Blink.
• Magi's Blessing was common for all roles.
• Hide of the Urchin is popular for anyone but Mages.
• Physical characters usually get Warrior Tabi, Jotunn's Wrath, and Titan's Bane. Other items (in descending order of popularity) include Deathbringer, Rage, Qin's Sais, Shifter's Shield, and Void Shield.
• Warriors (bruisers) don't pick purely offensive items other than Jotunn's Wrath, Titan's Bane, and Qin's Sais (Guan Yu gets not even this).
• As a sort of response to Rapitor's answer, I checked how many times Ninja Tabi appeared and noted only three instances: Anatoliy as Sun Wukong in round 3 of Dignitas vs. COG Gaming, GameHunter as Tyr in round 3 of TSM vs. Exposed, and InfernalTwist as Sun Wukong in the first round of Rage vs. Dignitas. Otherwise, Warrior Tabi was picked by an overwhelming majority.
• Mages usually get Shoes of the Magi, Rod of Tahuti, and Obsidian Shard. They often get defensive actives, like Aegis, Beads, and Combat Blink, but sometimes Heavenly Agility or Greater Sprint. Other items (in descending order of popularity) include Polynomicon, Shoes of Focus, Breastplate of Valor, Spear of the Magus, Bancroft's Talon, Chronos' Pendant, Void Stone, and Gem of Isolation (seen on Chang'e, Poseidon, and Ao Kuang.
• Guardians (tanks) usually get Sovereignty, Shoes of Focus, and Shell of Absorption.
• Chronos commonly gets Hastened Fatalis and less commonly Demonic Grip.
• Freya commonly gets Hastened Fatalis and Demonic Grip.
• Vamana commonly gets Hastened Fatalis.
• Demonic Grip is common with Freya but less common on Chronos, Poseidon, and Nu Wa (small sample size).
• Top Arena League player CeeVee usually gets Rod of Asclepius and Heavenly Agility as Chang'e.
• Greater Blink is common with Ares, Athena, Bacchus, Chaac, Geb, Hun Batz, Kumbhakarna, Odin, and Tyr.
• Be mindful that some items may be good with other characters unmentioned, but I haven't mentioned them because the sample that I studied (Alienware Arena Cup teams) did not pick those characters, which hints at which characters that you think would go good with an item that I mentioned are not good characters for Arena.
• A new item to consider is Ancile for physical characters, though it is a stacking item (+5/+1, 60). It maxes out at +50 Physical Power and +15% Cooldown Reduction (and +50 Magical Protection).

Some URLs redundantly appear as multiple sources in the code. If someone could clean them or teach me how to (efficiently), I'd appreciate it.

Something to note is that like any game mode and god, there is no end-all item build. You build what you need for the situation. They have extra tanky characters? use executioners. More mages? Stone of Gaia. All Items are viable and situational.

a few examples:

Warriors benefit much more from Ninja Tabi instead of Warrior (more cdr as warriors do more with their skills)

Mages like Ao go for stackable items (especially Book of thoth because he has no trouble farming minions, even in crowds.

Guardians lean more towards CDR with Breastplate of valor and Shoes of focus as it lets them control the battlefield more often.

Hunters like swapping out an item for either fatalis or Frostbound to make the wide open area of arena more manageable in combat.

Assassins also go for movement speed and/or the Hunter-mentioned weapons as it lets them cover more ground and chase better, as arena doesn't have the flank/gank potential of say Conquest or Siege.

Much like any of the other game modes, there are core items that are for that god regardless of mode (like fatalis on Vama/Freya), but listing each and every 'core' for each god is far too much for an answer.