Agatha Ripp - S2-E8
Audio problem: When Christian is returning to the office and sees a crowd of people outside hoping to see Agatha (who they believe has the stigmata of Jesus Christ), he runs into a couple with their infant. Christian asks if their baby is sick, and the couple shows him the backside of the infant where it has a vertebrae "tail", and they then ask him if they can see Agatha. During this sequence, we can hear loud crying, but various camera shots show the baby's face and she/she is not crying.
Other mistake: When Sean is on the phone with Julia, by mistake he accidentally calls Julia "Julian." Julian is the first name of the actor who plays Christian.
Factual error: At the onset of the episode, Christian and his blind girlfriend decide to go to a movie rental place. While Christian goes inside, he leaves his gf in the car and the song "Love Will Keep Us Together" by Captain and Tenille is playing. During this time, two young guys pull up in a van and find out the girl is blind and then decide to strip Christian's car of all 4 wheels and the front end. Christian comes out about three minutes later (the end of the same song is nearing) to see his car sitting on 4 jacks and void of any wheels or front end. Given the time that Christian is gone which is just a few minutes, it is nearly impossible for even a team of multiple seasoned pro thieves with the most sophisticated auto equipment to disassemble part of a car (and those specific parts) in that small amount of time, let alone two kids just passing by deciding to do so on the spur of the moment.
Plot hole: By the end of Season 3, it is revealed that Dr. Costa was the Carver. It is important to also note that the first introduction to the Carver came around mid-Season 2, and there were more than a handful of victims - including Dr. McNamara himself - all in the Miami area. One huge discrepancy during that initial time period though is that Dr. Costa was living in Atlanta, running his own plastic surgery firm (we know this because Julian flew him in from Atlanta to work on Dr. McNamara). While not impossible, it just doesn't seem logistical for Dr. Costa to commute back and forth from Atlanta to Miami to commit these crimes, all while still working/running his practice in Atlanta, especially in such short spurts of time.
Joy Kringle - S3-E13
Continuity mistake: Right at the part where Joy Cringle is prepped for surgery and Christian says "Okay, ready to suck the pringles out of Mrs. Cringle?", there is a close-up of Joy's face. Although she should be completely out, it is real clear her eyelids flinch a few times.
Cherry Peck - S3-E14
Continuity mistake: After Sean and Christian open the 'bleeding package' and discover Kimber's implants inside, Christian notices a very very small piece of paper inside one of the implants with a message on it (so small that he had to put it under a microscope to read it). A few minutes later in the episode, however, he shows the piece of paper to Sean, and it is significantly larger in size.
Plot hole: Towards the end of the episode, Mr. Alderman ties up Cherry Peck and forces Matt to cut off Cherry's male genetalia. Matt does so, and Cherry screams in pain. Later on that night (probably at least an hour later in what would be "real time"), Mr. Alderman and Matt drive to a secluded area and Mr. Alderman forces Matt to bury Cherry alive. Matt refuses, and Cherry eventually steps out of the grave and strikes Mr. Alderman with a shovel before shooting him. Cherry has the same clothes on, but is in no sign of pain, or is even bloodied from the earlier trauma. It is safe to assume that neither Matt nor especially Mr. Alderman would have driven Cherry to any hospital, nor would any hospital even have let her out in an hour or two's time after an injury like that.
Continuity mistake: At the climax of the episode when Quentin is revealed at the Carver, he has Sean and Christian tied up in their chairs and he cuts one of Sean's fingers off, of which Sean screams in pain. A few minutes later after Quentin is shot dead and officers/paramedics are assisting the scene, an overhead camera shot shows the entire area and floor around Sean's chair without any ounce of blood. Wouldn't there have been tons of blood from a finger being cut off?
Answer: "All I Know" by Art Garfunkel.