Thank You for Not Morphing - S1-E3
Visible crew/equipment: In the scene where Prue comes home and finds Andy inside the manor. There is a shot where if you look behind Andy's shoulder you can see half of the body of the director in the background in the observatory room sitting in a director's chair. (00:16:00)
Something Wicca This Way Comes - S1-E1
Continuity mistake: In the scene when Piper and Jeremy are in the taxi, Piper gives him a fortune cookie. You can see her closing the box. In the next shot of Jeremy, you can see the box is still open. (00:31:15)
The Truth Is Out There... and It Hurts - S1-E8
Revealing mistake: When Phoebe asks Piper what she thinks of her boss, you can see that Holly's tattoo has been covered up, because that part of her wrist is darker than her skin. (00:17:15)
Suggested correction: Maybe Holly covered it up with makeup because she isn't supposed to have tattoos for her job? A lot of jobs still don't allow piercings and tattoos today but all those years ago it was even harsher.
There are multiple instances of covered tattoos during the early seasons. I noticed it the most on Alyssa's back tattoo.
Love Hurts - S1-E21
Continuity mistake: When Prue shoots the Darklighter arrow at Alec, he orbs out and the arrow sticks into the wall behind him. In the next shot, the arrow has disappeared. (00:25:45)
I've Got You Under My Skin - S1-E2
Continuity mistake: When Phoebe flirts with Alec at Quake, she rests her head in her hand. In the following shots between Alec and Phoebe, her hand moves back and forth from her face to the tabletop. (00:02:00)
Wicca Envy - S1-E10
Continuity mistake: At the beginning of the episode when Piper hides her sisters from Leo she has the shirt with the buttons closed, but between that scene the shirt is not closed. (00:01:05)