The idea that Freeza's entire race is as naturally powerful as himself. While King Cold, Cooler, Chilled and Frost do have comparable power, Word of God says that Freeza is a mutant member of his race that, like his father, was born with abnormally huge power. There's also the idea that Frieza's entire line undergoes the same transformations, where only Freeza and Frost had forms highlighting their Bio-Armor, and Cooler alone has a fourth transformation. In particular, there's a recurring idea among some circles that King Cold can transform, though there is absolutely no evidence for this in any official material (not even What If?-obsessed media like Heroes).
Speaking of Freeza, given how murky or nonexistent his backstory is, fanon has inevitably attempted to fill in the gaps, regarding everything from the name of his race/species, to his actual age, to why he is apparently the only villain in Dragon Ball Z who is evil of his own free will.
Recurring fan theories involving Saiyans:
Saiyans mate for life.
Dragon Ball fanfiction is almost universally permeated with the notion that Saiyans undergo some sort of physiological/psychic connection when they fall in love, that Saiyans mate for life, refer to their love interests specifically as mates, and that the connection is formed in a process called "bonding" which involves one or both partners biting the other on the neck vampire-style. People may have originally gotten the idea for the psychic connection thing from a filler scene in the anime where Bulma gets a horrible feeling as Vegeta dies against Majin Buu. However, this is the only scene even close to suggesting this, appearing to show that numerous writers have blown it out of proportion.
It could also be said for when Videl senses that Gohan is alive despite everyone thinking he's dead.
Likewise in Super, there's an episode where Future!Mai seems to communicate with Future Trunks telepathically (he hears her voice despite her lips clearly not moving).
It's especially bizarre since we see in Dragon Ball that Master Roshi can read minds if he focuses. Saiyans, meanwhile, didn't even know ki sensing was possible so it doesn't seem likely they'd have been able to use telepathy all this time.
It's canon, at least for Goku; him reading Krillin's mind on Namek is in the manga. Goku can read minds (jury's out on the rest of the Saiyans), he just never remembers that.
Yamcha was fired from his baseball career for being in a fight. Not only is this never mentioned and was Yamcha explicitly holding back, but the characters indicate that fights breaking out were a common occurrence in the stadium (likely a reference to hockey riots). Yamcha is also seen playing after the Saiyan Saga and all the way into Super, disproving this.
Another example is that Yamcha was terrified of the Saibaman and only decided to face them after Tenshinhan because of a desire for glory. Not only does Yamcha never display any fear of the Saibaman, but he explicitly volunteers in place of Krillin because the latter had already died in the King Piccolo arc and thus couldn't be revived by Earth's Dragon Balls.
There seems to be a consensus among fans Chi-Chi is the reason Gohan stopped training in the seven years between Cell and Majin Buu Sagas. Gohan is the reason Gohan stopped training. He doesn't like fighting, period.
Extending to the above, Chi-Chi physically abusing Gohan with a frying pan when he didn't do what she wanted. She was strict with him, but never abusive.
Broly. Due to a liberal translation in the FUNi dub of the movie, he is almost universally thought to be increasing his power level at a constant rate, indefinitely, while in his Legendary Super Saiyan form. (The Dragon Ball Multiverse fan manga has actually made it a plot point.) That Broly replenishes his Ki and stamina and keeps them at the top no matter how long he fights is closer to the truth. Which, given the Saiyan ability to grow stronger every time they fight really could lead to power increases; not endlessly getting stronger, but close enough that no one can tell the difference.
The most commonly accepted explanation for the heart disease that took Goku out of action in the Android saga and killed him in Future Trunks' timeline is that he contracted it on Yardrat and brought it to Earth. Despite showing symptoms a year later in Trunks' timeline and three years later in the regular timeline, diseases lying dormant in a body for years is not unheard of, it's actually a Truth in Television. We're not given enough information on his heart disease to understand what it fully is but many fans assume it to be a virus of some sort because it fits the characteristics of virus latency. Also, if not a virus, there is the possibility of incubation period which is the period between infection and clinical onset of the disease. It's possible that training for the Androids gave Goku a healthier body which allowed his immune system to fight off the disease for a longer period of time but on the day of the Android attack fighting them overexerted his body and the pathogen struck. Which could be related to the French dub saying he got it on Namek (and yes, why then Vegeta, Krillin and co. didn't is anyone's guess).
Chi-Chi attacking Goku (or anyone else) with a frying pan, despite the fact that she was never seen doing this in the original series. However, in the 2008 special, Chichi was finally seen using a frying pan as a weapon on Master Roshi, making this official canon.
On the subject of Bardock, Seripa/Selypa/Celipa (the lone female of his team, named Fasha in the dub) was frequently singled out as the mother of Goku (and Raditz... sometimes) due to being the only named, full-blooded female Saiyan in the franchise for close to two decades. This is despite her relationship with Bardock not being portrayed as anything more than platonic/professional and, perhaps above all, Seripa being absent when Bardock interacts with his infant son. In fact, the special seems to imply that something's instead going on between Seripa and Toma (Tora). The introduction of Gine in Dragon Ball Minussquashed this idea, naturally.
Capsule Corp is usually portrayed in fan fiction as an economic giant with resources to rival a small country. While not inaccurate, this often carries the side effect of Bulma having fame to match post-Cell Games Mr. Satan. The closest the actual series gets to this is when the people of West City shrug off Porunga's summoning as another one of Capsule Corp's crazy experiments.
In the 2014 full color re-release of the manga, Akira Toriyama revealed that Android 16 was modeled off of Dr. Gero/Android 20's son, a Red Ribbon Army soldier who was killed by friendly fire. However, plenty of fans don't know or recognize that last fact and instead believe that Gero's son was killed by Goku during the events of the original series, which is why Dr. Gero wanted revenge against him. This mistaken impression is probably aided by the fact that in Dragon Ball Z Abridged, Goku did kill the younger Gero (as seen in a rather well-made original animation in Episode 56).
Another misconception is the idea that movie villain Bojack was sealed inside King Kai's planet and released when it was destroyed by Cell's attempt at Taking You with Me. This resulted from the original English dub making things up; the original Japanese version simply said that he'd been sealed away by the Kais, and King Kai getting killed by Cell released the seal. This is another one Abridged has contributed to, which has Bojack literally being trapped inside the planet and occasionally conversing with its inhabitants (to King Kai's annoyance) as a Running Gag.
Very few characters in the series have last names, in fact, the manga mentions last names have disappeared from use, yet fans insist on giving them last names. The biggest offenders are:
Bulma Briefs, though this misconception is caused by the dub. Bulma's father is called Dr. Brief and the dub assumed that was his last name because that's how it's usually done in real life. However, in the Dragon Ball universe, that's just his title followed by first name. The dub also called Bulma's mom "Mrs. Briefs", which cemented Brief(s) as a last name. She was just called "Bulma's Mom" in the manga and anime, though Toriyama said that if he'd given her a name it would have been "Panchy" note a play on "panty".
Videl Satan, another misconception also caused by the dub when Mr. Satan's name was censored as "Hercule." Thus, fans assumed his name is "Hercule Satan", when in reality, Hercule is a dub-only name, Mr. Satan is his ring name, and his actual name is revealed only in supplemental as Mark (rendered phonetically in Japanese as Maaku, an anagram of Akuma, which means "devil"—in line with Videl's own name being an anagram of devil).
He was called "Mr. Savage" on some of the earlier DBZ products before the Cell Saga premiered in the U.S. Savage (an actual real-life surname) could be his real last name (i.e. Mark Savage).
In fact, canonically the only characters in the show who do have last names are the Son family, our main characters Son Goku, Son Gohan and Son Goten. When Gohan went to high school, other students even pointed out how unusual it was that he had a last name. Also, of course, Son Gohan Sr. Contrary to popular belief, though, not Chi-Chi. She is never addressed that way in the series, and Chinese women do not change their last name in marriage.
Among English-speaking fans in particular, there's this notion that Gohan's appearance in the Cell Games left more of an imprint on the public consciousness than the other fighters, and he became known as "the delivery boy". Canonically, there's no indication he's any more remarked upon by history than anyone else who was present, who are all able to walk around in public pretty much incognito.
Before Brolywas integrated into the canon of the series, many fans believed Kale was his Alternate Self just like how Frost is one to Freeza, and she is the Legendary Super Saiyan. At least in the manga, she is considered the "Legendary Saiyan" or a "Demon Saiyan."
Along with this, for fans who didn't believe in the legend to begin with (since regular Super Saiyan is supposed to be already what the legend is like), a popular theory was that Kale and Broly were mutants to the Saiyan race like Freeza and his father were to theirs. Became Ascended Fanon in Dragon Ball Super: Broly, with King Vegeta calling Broly a sort of mutant, though it's not been explicitly said if the same applies for Kale.
In the manga, it is stated that the Saiyans from Universe 6 evolved differently from their counterparts, losing their tails in the process. Some fans like to theorize the Universe 6 Saiyans come from a long line of mixed relationships, very similar to how Goten and Trunks are half-human, half-Saiyan breeds, and this explains why they are so powerful, something that isn't exactly stated.
Akira Toriyama's interview about the S-Cells actually state there is a greater chance for Saiyans to go Super Saiyan in a healthy environment like Earth, which could add a little fire on this theory.
A lot of fans believe that thanks to having the DNA of Freeza and the Saiyans, Cell could actually obtain his own version of the Golden Frieza transformation from Dragon Ball Z: Resurrection 'F', along with godly ki, with only a few months of training. Cue to Fanfic Fuel about Resurrection 'C'.
Fans like to speculate that Android 17's wife is L(a)unch, a character infamous for undergoing Chuck Cunningham Syndrome in-between the original Dragon Ball and Z.
It's popular in the fanbase to pair up Hit with the Universe 6 Saiyans, with him as a sort of mentor to them, in some cases similar to what Piccolo is to Gohan. While in Episode 111 of Dragon Ball Super, Hit does encourage them to keep with their work to fight and win in the tournament, and they appear together in the out-of-continuitySuper Dragon Ball Heroes, their relationship seems to be strictly professional; Cabba respects him and Caulifla (as Kefla) wants to fight him in the video games, but that's the extent of their relationship.
The Super Saiyan 4 form name, for the fans who saw Broly's legendary transformation being adapted and being named "Berserker Super Saiyan", has earned the Fan Nickname of "Super Saiyan Feral", among others. It helps the form is not really an evolution of Super Saiyan 3 and does not resemble any previous Super Saiyan transformations.
Also, if not a virus, there is the possibility of incubation period