I really feel like there isn't enough discussion about ships in the Avatar universe yet so I decided to list my own here and encourage others to do the same. Please note that my Legend of Korra ships and Kyoshi novels ships will be in separate posts.
Aang/On Ji (On Jaang)-I'm not sure what drew me to this ship. Maybe the fact that On Ji liked Aang even though she didn't know he was the Avatar and was among a group of people who saw him as a delinquent. I've talked about it extensively in this post.
Aang/Toph (Taang)-I know it seems like this pairing only exists for Zutara but that is only partially correct. I go so much more in-depth on this post.
Bato/Hakoda (Bakoda aka Water Tribe Husbands)-I mainly shipped this before North and South introduced Malina. Hakoda was already a widower before the series even started and other Avatar fans convinced me that they would be a good couple.
Bumi/Herbalist (Bumerablist)-I came across this pairing on Avatar Wiki's Fanon:Shippings page but I don't think that anybody actually shipped it because it's just a crack pairing. I started shipping it after thinking about how Bumi is a mad genius and how the Herbalist flat out admitted to being insane. They would definitely hit it off if they met.
Cabbage merchant/Cabbages (My Cabbages!)-Because why not?
Chan/Azula (Chanzula)-I mainly ship it because of their fling and having someone to pair Azula with. If only Azula didn't wreck Chan's house.
Chong/Lily (Chily)-We are talking about two hippies in love here. What isn't there to love?
Gansu/Sela (Gansela)-Pretty much because it's canon. They're that farmer couple seen in Zuko Alone.
Hakoda/Kya (Hakya)-We wouldn't have Sokka or Katara without them.
Hakoda/Malina (Halina)-Seeing how well they got along in North and South and Malina's near-death just tugged my heartstrings.
Haru/Katara (Harutara)-I guess I just like that it's the rarest of the 'four main' Katara pairings and how good Haru is to Katara.
Haru/Ty Lee (Ty Laru)-You can blame School Time Shipping for that one. I wouldn't have thought about this pairing otherwise.
Ikem/Ursa (Urkem)-They were meant to be together and even the royal family of the Fire Nation couldn't keep them apart forever. I'm really glad that they got their happy ending.
Iroh/Lo (Loroh)-It's crack and a shipping panel introduced me to them.
Iroh/Ticket lady (?)-They flirted in an episode. That's really all there is to them.
Jet/Mai (Jai)-It's another crack ship that the fans have introduced me to. I can imagine an amnesiac Jet finding Mai after his near-death and Mai's breakup with Zuko and them falling for each other.
Jinzuk/Rina (Rinzuk?)-We wouldn't have Zuko or Azula without them.
Kei Lo/Mai (Mai Lo?)-I just liked the idea of Mai moving on and really hoped that they were heading in that direction.
Kuruk/Ummi (Kummi)-One of the tragic love stories of A:TLA and I love the theory of Ummi being reincarnated as Toph.
Lao/Poppy (Laoppy)-We wouldn't have Toph or any of her wonderful children or grandchildren without them.
Longshot/Smellerbee (Smellershot)-I thought that this pairing was overrated at first but I could see the subtext after re-watching the series. I don't know how I missed it, to be honest. It just feels so obvious.
Momo/Appa (Moappa)-This was mainly a joke ship to me but the two really do like each other a lot.
Ozai/Zhao (Zhaozai)-Both of those men are such jerks that they really don't deserve anybody else but each other.
Pakku/Kanna (Kakku)-It became canon in the finale. I was so depressed over Zutara not happening that another fan had to remind that some good came out of the finale too. Old people deserve love too.
Roku/Ta Min (Ra Min)-Another wholesome ship involving the Avatar that we would not have Zuko without.
Sokka/Suki (Sukka)-Suki really did bring out the best in Sokka. She really did.
Sokka/Yue (Yukka)-Sokka's first girlfriend turned into the moon. It was really rough, buddy.
Than/Ying (Thing?)-The refugees traveling as they were getting ready to have their baby just tugged at my heartstrings.
The Duke/Toph (The Doph)-I don't know why but when it was first announced that Toph had a daughter I really wanted the father to be The Duke. We already 100% knew that Taang was a no-go at this point.
Zuko/Jin (Jinko)-This is the pairing that I am pulling for at this point since we know that Zutara is not canon.
Zuko/Katara (Zutara)-I could probaby write a whole book about them if you really wanted me to. I remember the good old days when "Zutara Evidence for the Intellectual Dummy" was around and convinced me to go for this ship. You can disprove it a thousand times but I still know that they are good for each other. And in The Legend of Korra they are both still alive so it could still happen.
If you want to know my opinion about a ship that is not on my list let me know and I'll tell you. I also invite other Avatar fans to tell me about their Avatar ships.