Visible crew/equipment: The stage lights are reflected on the vase in the family room. (The one with the pink flower design).
The Fresh Prince Project - S1-E1
Continuity mistake: When Will puts the last of his wall-shaking staples on the Malcolm X poster before the butler's interruption, the poster is almost perfectly flat. When Geoffrey walks to the poster, a bottom corner is noticeably raised. When Carlton walks to it, it's flat again, and raised when Will freaks out at the mirror. (00:07:40 - 00:10:00)
The Fresh Prince Project - S1-E1
Revealing mistake: When Will screams in front of the mirror, imagining himself turned into Carlton, something went wrong making the composite shot and not only you can actually see the line of separation between the two frames, but the superimposed one exposes a bit of the mirror woodframe before it wiggles into the correct position and covers it up. (00:11:30)
The Fresh Prince Project - S1-E1
Revealing mistake: When Will staples his Malcom X poster to the wall, the wall bends back and forth for a second or two.
The Fresh Prince Project - S1-E1
Visible crew/equipment: During Will's first introduction to Uncle Phil the camera tilts slightly down to Will and Aunt Vivian on the couch and the boom mic drops into the frame briefly.
Continuity mistake: When Ashley shows Will her weekly schedule, her hand is towards the middle of the upper roll of the parchment, but in the reverse shot where the calendar can be read, her hand is at the top right corner. (00:03:30)
Revealing mistake: When Phil is looking for Will who is in Ashley's room he slams one door shut and the set wall can be seen to wobble. (00:11:00)
Character mistake: When Geoffrey serves tea after his sassy cough at Vivian's sentence about High Tea not being a sacred institution, the actor messes up when it's time to actually serve tea, with a napkin getting caught in the teacup saucer. Granted, accidents do happen, but it's quite uncharacteristic for the perfect butler Geoffrey is portrayed to be, and simply a flub by Joseph Marcell. (00:17:45)
Continuity mistake: When Phil is telling Geoffrey about his day at work he takes an aspirin and drinks the water. The glass of water is almost gone. But if you look closely a couple of shots later the glass is full of water again.
Plot hole: Ashley says she is 9; later in that season, in "Just Infatuation," Phil says she is almost 12. Will is 17 for the first two seasons, and 18 in season three. Nicky Banks grows from baby to preschooler between seasons four and five (though this was humorously addressed).
Clubba Hubba - S1-E3
Continuity mistake: As Carlton and Will battle out with their "I bet he won't", "I bet he will", Will is holding his drink with his left hand. When he breaks the fourth wall with a joke in a new shot, the glass switches hands. (00:05:00)
Clubba Hubba - S1-E3
Revealing mistake: During his first conversation with Dr. No, Will Smith mouths "strings of Arabians" as Richard Roundtree says it (it can't be a line he rehearsed with Carlton, because his character has no clue about that branch of conversation). (00:11:00)
Not With My Pig, You Don't - S1-E4
Continuity mistake: During the award dinner, when Hattie says the line about "Zeke sprung from the ground like a big ol eggplant," watch the table in the back, next to the door. The man sitting there wipes his mouth raising a napkin 3 times in 3 separate shots. (00:19:00)
Homeboy, Sweet Homeboy - S1-E5
Continuity mistake: When Ice Tray announces that Hilary and him got married, Mrs. Banks' glass is suddenly emptier, going back to almost full just a moment later. (00:17:45)
Continuity mistake: When the family kisses Will goodbye and when he sits bummed out on the couch, the pool float around his arm switches position, going under his left arm, then under his right, then left again. (00:02:50)
Audio problem: When Jerry goes over to open the jail cell door, the sound effect of jingling keys can be heard, however the actor isn't holding any keys. The jail cell door doesn't even have a keyhole.
Suggested correction: When the Sergeant tells Jerry to open the cell (00:18:20), Jerry reaches to his left side to retrieve keys from his duty belt (there is no "sound effect of jingling keys"). And keys were visible dangling from Jerry's duty belt when he told Philip to calm his wife (00:17:10). Secondly, the jail cell door does indeed have an old-style keyhole (00:18:30).
Continuity mistake: A member of the poetry club responds to Will's "You're a poet and you don't know it!" with a sassy "I -AM - a poet and I -DO - know it" in a close-up. In said close-up, she is holding her hand on a book, while in the shots before and after, her arm is in a different position. (00:06:30)
Continuity mistake: In the scene where Will tells Vivian and Phil that Carlton is in Compton, Vivian takes off her earring from her right ear. A few shots of Will pass and when she goes to take off her other earring, the left ear doesn't have one, and the earring is on the right ear again.
Did the Earth Move For You? - S2-E1
Other mistake: Viv and Phil are waiting in his office for the power to come back on after the earthquake, but you can see lights on in the building across the street. (00:16:30)
Did the Earth Move For You? - S2-E1
Other mistake: Ashley is now thirteen years old, but she was nine in the previous season.
Suggested correction: It's not uncommon for TV shows to do this, especially with kids. In this exact show they had a baby Nicky who the following season suddenly was like 5 or 6 years old. Other shows like Family Ties and Growing Pains did the same thing. Shows are not always shown in real time when it comes to episodes. They can each take place days, weeks, months or years from previous episodes.
This isn't a valid correction. Just because shows do it or it's common doesn't invalidate the mistake. And while shows don't always have to be shown "in real time", all the other characters would have to make the same jump in age, otherwise it's a mistake.
As Bishop73 said, it's a legitimate mistake because the other characters did not age by four years. There was no time jump. Changing her age does not make sense within the "world" of the show.