Dragon Ball Z 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 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.
Broly. Due to a liberal translation in the FUNi dub, he is almost universally thought to be increasing his power level at a constant rate, indefinitely, while in his Legendary Super Saiyan form. (The DBZ Multiverse fan manga has actually made it a plot point). Actually that Broly replenishes his Ki and stamina and keeps them at the top no matter how long he fights is closer to the truth.
A small but vocal section of the fandom believes that male Saiyans are capable of getting pregnant and bearing children, due to the fact that nearly all of the Saiyan characters that have showed up in the series are male. This may have been helped by the fact that, before Caulifla and Kale appeared in Super (and they're from an alternate universe where Saiyans went on a very different route), the only female Saiyans who appeared on the series were a member of Bardock's crew from the Episode of Bardock special, and Gine, Goku's mother, from the Dragon Ball Minus manga.
A different portion takes the same evidence and comes to the conclusion that Saiyans have a higher sex ratio than humans, such that male births are more common than female births.
Another, less disturbing bit of fanon is the general acceptance that the heart disease that took Goku out of action in the Android saga, leading to the havoc on the world in the alternate timeline was caught by Goku on Yardrat and brought to Earth. Despite contracting it 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 officially canon.
Mystic Gohan. While this is a form that unquestionably exists, being the one unlocked by Old Kai in the Buu saga, that specific title was never used in the official material (licensed games referred to this form as "Ultimate Gohan").
Videl states that surnames have fallen out of fashion in the world where Dragon Ball takes place, to the point where she guesses that Son Gohan is the son of Son Goku because very few families still use surnames. Nonetheless, it's common for fans to give characters surnames, for example: Bulma Briefs and Videl Satan, despite those being Bulma's father's given name and Videl's father's stage name, respectively.
A widely accepted fanfiction idea is Vegeta was raped/abused by Frieza at some point during his time working with him, and the abuse/rapes were what fueled his treachery.
Due to how Goku and Vegeta don't seem to age between the beginning of Dragon Ball Z and the beginning of Dragon Ball GT despite how many years pass, many fans have concluded that Saiyans age slower and have longer lifespans than normal humans. Like the Mister Seahorse theory above, this often shows up in Goku/Vegeta yaoi fics, usually as a way to get rid of Chi-Chi and Bulma.
It is presented as canon that Saiyans age slower than humans and stay in their physical prime longer. This is explicitly stated by Vegeta after the post-Buu timeskip in Dragon Ball Z, before the final tournament. Whether or not they have longer lifespans is up in the air.
Assuming a Saiyan's body works similar to a human's (and considering that Saiyans and humans can interbreed with no real medical intervention it's a safe assumption) one can assume that slower ageing and a longer physical prime would translate to longer lifespan because aging is actually a result of the body's natural ability to regenerate and repair itself deteriorating. The older we are the less the body can regenerate resulting in things like slower healing, weaker immune systems, and the various physical signs of ageing (impaired senses, difficulty moving, wrinkles and graying hair, etc). Most old age deaths are either from the body not being able to recover from illness due to a weakened immune system or just the body shutting down because it can't support itself anymore. If Saiyans age slower and have a longer physical prime, that means their bodies hold onto their natural regenerative abilities for longer. Therefore unless Saiyans abruptly hit the wall and go from their primes to decrep in 10 years they must have a longer lifespan.
A lot of fan fictions tend to have Vegeta never calling someone by their name aside from Goku and Trunks, like "Earth Woman" for Bulma, "Kakarot's" son for Gohan, or "The Bald One" for Krillin despite the fact that Vegeta regularly call them by their name, particularly after his Character Development.
Capsule Corp is usually portrayed in fanfiction 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.
From Dragon Ball Super
Even if Broly doesn't exist in the main canon of the series, many fans believe and like to think Kale is his Alternate Self just like how Frost is one to Frieza, and she is the Legendary Super Saiyan from Universe 6.
Adding to this, to the fans who don't believe in such legend being adapted (As Kale's form is called Berserker), some people believe she (and possibly Broly) is a mutant to the Saiyans, similar to how Frieza is stated to be one for his race.
In the manga, is stated the Saiyans from Universe 6 evolved in a different way 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.
Toriyama 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 Frieza and the Saiyans, Cell could actually obtain his own version of the Golden Frieza transformation, along 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.
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 the previous Super Saiyan transformations at all.