DISCLAIMER: MOST OF THE CONTENT IN ESO DOES NOT REQUIRE BEST IN SLOT GEAR, AND EVEN SOME VERY DIFFICULT CONTENT CAN BE ACCOMPLISHED WITHOUT A FULLY OPTIMIZED SETUP. THERE ARE MANY "GOOD" SETS AVAILABLE THAT WILL SERVE YOU WELL. FEEL FREE TO USE WHATEVER FEELS ATTRACTIVE TO YOU AND IN MOST CASES YOU'LL BE FINE! 
With all the new gear sets available in One Tamriel, many players new and old are trying to figure out which sets will be most beneficial for them. There are some notable build creators such as
@Alcast and
@Deltia who post in-depth build videos on Youtube that are definitely worth looking into, but what I would like to do in this thread is simply compile community opinions about what gear is generally considered to be BiS (Best in Slot) for all standard classes and play styles. Differences of opinion are welcome! People can use the "agree" button to upvote gear sets that are mentioned. You can also mention sets that you don't consider BiS but are still very good and worth having for certain circumstances. Although other factors like skill choices, traits, and enchantments are important features to a build, I would like to keep this thread strictly focused on the gear sets themselves. My hope is to help people feel more confident about what sets they choose to keep and what they choose to decon. Even if you don't have a DK tank yet, maybe you will want to make one later. Why not know what gear would be good for that now and avoid deconing something you'd wish you still had?
You can submit a single gear set (like Ilambris or Twice-Born Star) or a full gear setup. Please indicate which category you are submitting for. I will update the OP to reflect feedback received. Also, if you are pulling your info from someone else's build, please do give them credit when mentioning it! You can also provide the link to a build page or video if there is one. For myself, I main a PVE Mag Sorc and am totally on the "Mystic" bandwagon from
@Alcast. My knowledge of most of the other classes and specs is not large, which is one of the main reasons I'd like this to be a community project. So let me hear from you!
EDIT: Added a Maelstrom category. This gear set will be considered BiS for Solo play, while the DPS, Tank, and Heal categories envision participation in Trials groups. 4-player content could potentially be completed with either type of set.
DRAGONKNIGHT (Stamina)PVE
---DPS: 2 Kragh + 5 Leviathan or 5 Twice Born Star + 3 Vicious Serpent + Maelstrom Dual Weild + Maelstrom Bow
http://alcasthq.com/venom/
---Tank: 2 Bloodspawn + 5 Alkosh or 5 Ebon (2 Body + 3 Jewelry) + 5 Tava's Favor (3 Body + 2 Sword/Shield [both bars])
http://alcasthq.com/catalyst/
---Heal
---Maelstrom
PVP
---DPS: 2 Tremorscale + 5 Black Rose + 5 Draugr (3 Jewelry + Dual Wield) + Any 2H
http://alcasthq.com/draugr/
---Tank
---Heal
DRAGONKNIGHT (Magicka)PVE
---DPS: 2 Grothdarr + 5 Burning Spell Weave (Deltia) or 5 Mothers Sorrow (Alcast) + 3 Infallible Aether + 2 Maelstrom Destros
http://deltiasgaming.com/eso-magic-dragonknight-dps-build-2/#end http://alcasthq.com/valakas/
---Tank
---Heal
---Maelstrom: 2 Engine Guardian + 5 Julianos + 3 Willpower + Maelstrom Destro + Malestrom Resto
http://deltiasgaming.com/eso-magic-dragonknight-dps-build-2/#maelstromPVP
---DPS
---Tank
---Heal
NIGHTBLADE (Stamina)PVE
---DPS: 2 Kragh + 5 Leviathan or Twice Born Star + 3 Alkosh or Vicious Ophidian + Maelstrom Dual Weild + Maelstrom Bow
http://alcasthq.com/rampage/
---Tank
---Heal
---Maelstrom: 2 Velidreth + 5 Night Mother's Gaze + 5 Vicious Ophidian + Maelstrom Bow
http://deltiasgaming.com/eso-stamina-nightblade-dps-build/#SoloPVP
---DPS
---Tank
---Heal
NIGHTBLADE (Magicka)PVE
---DPS: 2 Grothdarr or 2 Valkyn Skoria + 5 Twice Born Star + 5 Moondancer or 5 Spinners Garments
http://alcasthq.com/azure/
---Tank: 2 Swarm Mother + 5 Tava's Favor + 5 Spell Power Cure
http://deltiasgaming.com/eso-nightblade-tank-build-2/
---Heal
---Maelstrom
PVP
---DPS
---Tank
---Heal
SORCERER (Stamina)PVE
---DPS: 2 Kragh or 2 Stormfist + 5 Leviathan or Twice-Born Star + 3 Vicious Orphidian (alternately, no monster set, 5 Alkosh (2 Body + 3 Jewelry) + Maelstrom Dual Wield + Maelstrom Bow
http://alcasthq.com/windwalker/
---Tank
---Heal
---Maelstrom: 2 Velidreth + 5 Night Mother's Gaze + 3 Vicious Ophidian + 2 Maelstrom Daggers + 1 Maelstrom Bow
http://deltiasgaming.com/eso-stamina-sorcerer-dps-build/#SoloPVP
---DPS: 2 Tremorscale + 5 Black Rose + 5 Viper's Sting (3 Jewelry + Sword and Shield) + Maelstrom 2H Axe
https://www.youtube.com/watch?v=S1AgzidI4rU
---Tank
---Heal
SORCERER (Magicka)PVE
---DPS: 2 Ilambris + 5 Twice Born Star + 4 Infallible Aether or Moondancer (3 Jewelry + Destro) + Maelstrom Destro
http://alcasthq.com/mystic/
---Tank
---Heal
---Maelstrom: 2 Ilambris + 5 Julianos + 5 Elegant (3 Jewelry + Dual Wield) + Maelstrom Destro
https://www.youtube.com/watch?v=lf_hv5j0PqYPVP
---DPS: 2 Infernal Guardian+ 5 Spinner's (including destro staff) + 5 Lich (including resto staff)
https://www.youtube.com/watch?v=flBT7lSqmWM
--Tank
---Heal
TEMPLAR (Stamina)PVE
---DPS: 2 Kragh + 5 Night Mother's Gaze + 5 Alkosh (3 Jewelry + Dual Wield) + Maelstrom Bow
http://alcasthq.com/jabsmania/
---Tank
---Heal
---Maelstrom
PVP
---DPS
---Tank
---Heal
TEMPLAR (Magicka)PVE
---DPS: 2 Grothdarr + 5 Mother's Sorrow or Twice Born Star + 5 Infallible Aether or (3 Infallible Aether + 2 Willpower Dual Wield + Maelstrom Destro)
http://alcasthq.com/beamplar/
---Tank
---Heal
---Maelstrom
PVP
---DPS
---Tank
---Heal
GENERAL THOUGHTS
PVE TRAITS - Divines is best for all DPS and Heals, Tanks benefit from a mix of Reinforced, Sturdy, and Infused.
PVP TRAITS - Impenetrable is most important if you don't always have shields up. Otherwise, use PVE traits.
STAM BUILDS - Velidreth performs better in AOE and mobile play, Kragh performs better in stationary single target play.
https://www.youtube.com/watch?v=T4BfDAO5di0
CRIT SETS VS TBS -
@Asayre disputes
@Alcast in the use of Mother's Sorrow (and by implication Leviathan). You must have at least 66 points in Elfborn to make Mother's Sorrow out-perform TBS, and the cost of so many CP points plus the loss of additional Health and Stamina make TBS seem more optimal.