Who is the most useless character in DBZ?

Yamcha. I know he doesn't fit the criteria of "appears to have an important role" but the guy is completely useless. In DBZ his notable achievements included:

- Getting killed by a Saibaman
- Almost getting killed by #20 instantly
- Carrying Goku away after he has heart troubles
- Getting beaten by a Cell Jr (though to be fair, everyone was)
- Attending the tournament in the Buu-saga as a spectator

EDIT: Technically we can count RoF, since that was a DBZ movie initially before DBS was a thing.

- In RoF, Yamcha was so useless that the group decided Roshi would be a better replacement for him to fight against Frieza's army, and Roshi hasn't fought since the original DB.

Yajirobe. He may appear to have the important role of helping our Z Warriors get Sensu Beans, but honestly Korin is the real provider of them and the Z Warriors don't really need him there to deliver them every so often. I mean with Goku knowing Instant Transmission, he could always go get them from Korin himself and be back to the battlefield in a flash. Oh and yeah he managed to cut off Vegeta's tail, but to be honest almost anyone else could've done that.

Don't let Dr. Wheelo hear you dissing on Yajirobe. That guy was manly enough to survive Tambourine's rampage when Krillin didn't, and killed Cymbal. As far as Dragon Ball tiers go, Yajirobe was one of the top three strongest non-Kami good guys in the world come King Piccolo saga.

Probably Chaozu. He was set up as one of the main rivals, but he never really did anything. He was killed right after the debut arc, while Krillin was the first one to die he had two great showing in 21st and 22nd World Tournaments and his help in Red Ribbon arcs. Ultimately he was the one of his friends not to making it past the primarily round of the 23rd tournament and was absent for most of it, whereas Yamecha at least always made it to the main event and usually lose to the future winner of the tournament (Roshi and Tien) or you know god. Even his unique telekinesis, which could have been built upon was made irrelevant in the Saiyan arc. His personality was never fleshed out either. Outside of Tien he barely is allowed to interact with the other characters. Unlike Krillin and Yamecha who at least got decent development in DB so he comes off as just being there.    

King Cold. Even pathetic fighters like Chaiotzu and Guldo got to showcase their powers a little bit.
This twat got introduced just so Trunks could cleave him a new one.

Okay, this is actually a more complex than most people are giving credit, because there are many factors to consider.

Role:
What is this character's role in the story? a fighter? a support character? a villain? a gag character? I can't exactly blame Turtle for not throwing it down with Freeza, can I?

Attempt:
While it's true that you can't fail if you never try, you can't succeed either, so how hard has this character tried to keep going? did he or she try despite all odds?

Impact:
Quality over quantity, it doesn't matter if you did one good thing the entire franchise, you did something, you are better off than most.

Endearment:
Is this character important to the fandom? is he or she hated or beloved? even if a character never impacted the story, if they are beloved by the fans, they enhanced the franchise somehow, so the aren't exactly useless.

With all that said, let me go through some of the mentioned names...

Pilaf: The dude got awfully close to conquering the world, if he was a little more decisive about it and hadn't stuttered, he would've won, that's sufficient as far as a villain goes, his continuation in the story is dubious, specially now in Super, but... eh, I don't think Pilaf has ever got aggravating, even during Super.

Puar: He is a satellite character to Yamcha... but he does serve an important role in cutting off Goku's tail during his first trasformation into Oozaru, along with Oolong, his transformation ability has also served as a source of visual gags at times, so that's another thing he has going for him.

Oolong: He twarted Pilaf's attempt to conquer the world, and given that Pilaf was the Big Bad at the time, that's more than we can say about most characters in the franchise.

Yamcha: This guy never gives up, really, he was widely beloved during the early DB days, he never did much, but he did enough, honestly, TFS encapsulates Yamcha's character with a joke during the Cell Games, where he mentioned he brought towels and water bottles for the others, and that's him, he knows he has a -1% chance of doing anything important, but he will be there to give any support he can to his friends, no matter how small, and he IS one of Goku's closest friends, so he got that going on for him.

Chiaotzu: He... he... shit, what did he even do in the franchise...? only thing I remember him doing is dying, twice.

Yajirobe: Come on, he saved Goku's life, at a time where there were no Dragon Balls at that, that would be enough for most people, then he cut Vegeta's tail, which essentially saved the planet, given how weak Yajirobe is, he just did so much.

Shin: He was trying, he was trying so hard, and his situation is really not that good, he was left with trying to figure out what the fuck a Kaioshin was supposed to do on his own, so it's no wonder he haven't accomplished much, it's not until Elder Kai gets freed that he gets a proper mentor, and from the point Elder Kai is freed onwards, he just does his best to follow his lead.

So at the end of the day... the most useless character is... either Cold or Chiaotzu

Cold: His role was to be the victim of the Worf Effect in favor of Future Trunks, sure, but that's something we can tell in hindsight, when he was introduced, he seemed like a legitimate threat... and then he accomplished nothing, you can't even argue that Trunks needed the hype, killing Freeza was way more than enough, and given how debatable it is if he is supposed to be weaker or stronger than Freeza, with most people leaning towards "weaker", killing him after having killed Freeza isn't that impressive, that's like cleaning up the minions after you killed the boss.

Chiaotzu: I can't even remember what he did... I remember he actually has a much bigger role in the DB movie where he is a prince and Tenshin is his bodyguard, and that he is more important in a non-canon movie than in the actual story just highlights how largely unimportant he is, it's particularly bad when you consider he was supposed to be a foil to Krillin, and the Krill Master went on to be major player all the way to the Android saga despite the power discrepancy, I can't even remember he particularly tried, instead just fading more and more into the background, I sort of remember he always played a major moral support role to Tenshin, but while widely beloved, Tenshin isn't a particularly important character either.

Well chiaotzu was a dangerous opponent for krillin in the tournament. He also provided some comic relief. Later on he was mostly there to add some character development for Tien, to highlight a friendship and add some emotional depth to tien when chiaotzu sacrificed himself on nappa. After the tournament he was pretty worthless as a fighter and since Tien was also fading from the spotlight chiaotzu was more or less nixed from the story not long after so I'd say he was useful during the time he was around.

If you consider duration then cold was very useful for the very brief time he was around. What did he do? He showed us all how much of a no nonsense badass Trunks is and in stark contrast to goku and even vegeta to some extent. That's all he was there for and he did a great job at it.

My memory is a bit spotty on this but if we're counting the anime then I think grand kai was pretty damn useless as well. He was the greatest of gods up to that point in the story and as I recall all he did was preside over the tournament and when it was time to reward the winner he had nothing to give.

When you look at just how useful king kai was and how he was the lower god it really makes supreme and grand look awfully useless in comparison.

See, this is what I meant by "role" and "endearment", Launch didn't do jack, but she was never supposed to, that was never her role, her role was to be a two-fold character to serve Toriyama's interest in women, either a very calm, very gentle woman that fits Toriyama's views of a perfect wife since he appears to have rather conservative opinions about women(not criticizing him about his conservative views, he is an old dude, it makes sense he has them, but it is a fact they are there, and it's very obvious they are), while also being an aggressive, intense woman who is also very attractive, but in a different, more sexual and raunchy manner.

Turtle is equally useless, what did Turtle do? while he served as a means to introduce Roshi to Goku and give Goku the Nimbus, the fact stands that Roshi had a Dragon Ball, and Goku would have met him anyway thanks to the Dragon Radar, Roshi would have recognized the Power Pole as being Gohan's, and the same relationship between Roshi and Goku would have flourished, the one thing that Turtle does in the franchise, you can write around very easily without making it contrived in any way, but the thing is, that was never Turtle's role, Turtle was meant to be a minor character who served as to explain with who Roshi talks with.

And the applies to Launch, she was meant to be there for a few gags and light fanservice, she served her role and is a very popular character because her personality was interesting and fun enough for people just her mere presence, she left an impact in the franchise even without doing anything important, but she was never supposed to, and that's fine, the issue with "useless" characters, in my opinion, come from characters that either were meant to serve a bigger role and fall flat, or are stated as large and meaningful but end up doing jack because their role in the narrative was something that got in odds with their stated status in the story.