EDITED ENGLISH VERSION: In the original Japanese version, Dr. Briefs (Bulma's dad) ALWAYS had a cigarette in his mouth. It was digitally removed for the edited dub, but the editing is often unconvincing. Watch his mouth and you can see "glitches" on the screen in that area.
English dubbed version: Early in the series, Raditz states that the planet Vegeta was destroyed three years prior to the beginning of the series. Later on, this is changed to twenty years (which is accurate to the original Japanese version). When they went back and did the "uncut dubbed" version that aired on Cartoon Network, they still used the inaccurate "three years" quote.
(English dubbed version) In the episodes where Goku is training on King Kai's planet, it takes him six days to catch Bubbles. But in the Frieza saga when Goku's dead friends arrive there to train, King Kai tells them it took Goku three weeks to catch Bubbles.
In the Saiyan Saga, when Tien, Chiaotzu, Krillin, Yamcha, and Yajirobe first arrive at Kami's lookout, they have "flashbacks" of themselves in combat as Kami speaks to them. However, the events in these flashbacks have not happened yet (they would happen right after they started their training.)
In the Frieza saga, Vegeta has flashbacks both of himself with Guldo, and with Nappa and Raditz. In both of these flashbacks he is missing his tail, even though he didn't lose it until his battles on Earth.
Tien's failed suicide attack cracks Nappa's armor. When Nappa charges at the Z-Fighters (when Vegeta orders him to stop) his armor is in pristine, undamaged condition. In the scenes that follow it's broken again.
The popular line where Nappa asks what Goku's power level is and Vegeta screams "Its over 9000!" and crushes his scouter in the English Dub is actually incorrect. In the Japanese Manga he screams that it is over 8000, not 9000.
There are numerous animation errors and consistencies throughout the series. Often, for a single frame or two, someone will be drawn in the wrong outfit or their outfit will be the wrong color. Likewise, some characters general appearances/color schemes will change for no apparent reason (Nappa and Vegeta before their arrival on earth come to mind.)
Prior to Nappa and Vegeta's arrival on earth, both of them are colored wrongly. Vegeta has red hair and lime green armor, and Nappa has generic black Saiyan-style armor. When they arrive on earth, they are the right colors.
On the hat Krillin wears en route to Namek, Krillin is misspelled as "Kulilin." Kulilin is Krillin's name in the original Japanese Version. This is because l's and r's are commonly confused when translating from Japanese.
In the Majin Buu saga, Shen is well aware that two henchmen joined the martial arts tournament to drain the energy from someone extremely powerful, who ends up being Gohan. Shen allows this to be able to follow the henchmen and find their spaceship, where Majin Buu will be resurrected. However, if he would have impeded Gohan's energy drain in the first place, or in fact just even told the Z Fighters of everything before the henchmen attacked, the whole Majin Buu saga would be non-existent, as Majin Buu would not be able to be resurrected.
In one of the episodes when Vegeta, Gohan, and Krillin put Goku in the recovery chamber in Frieza's ship in the Frieza Saga, Vegeta says "By typing into this system, we should have Goku up to speed in no time." when he's supposed to call him "Kakarot" since both Goku and Vegeta were from Planet Vegeta before it got destroyed by Frieza. All of the Saiyans from Planet Vegeta called him "Kakarot" since "Goku" wasn't his real name.