Just before Freeman and Random beam up from Beta III, Ransom is holding a PADD in front of him, with his arms crossed. But when he materializes, his hands are folded behind him and the PADD is gone.
The position of the Starfleet delta emblem on Peanut Hamper's shell changes constantly between shots.
When Riker greets Freeman in the bar, the positions of his hands change from shot to shot during the exchange.
When Ransom is making his opening log, he's looking at a picture of Kirk and Spock standing in front of Landru, presumably when the former defeated the computer in the 23rd century. The two officers are wearing their tan and blue uniform tunics.
However, when Kirk defeated Landru ("The Return of the Archons"), he was wearing indigenous clothing.
However, when Kirk defeated Landru ("The Return of the Archons"), he was wearing indigenous clothing.
When the Pakled ship snags the Cerritos, it attaches two clamps to the front of the saucer section. When the shot changes, the clamps are now on opposite sides of the saucer.
When Rutherford accidentally reveals a new button on his implant, he looks up and sees it on the side of his head, above where his ear had been. But his eyes are still in the front of his head, so there's no way he could see this new button.
When Ransom and Freeman return to the bridge, no one is in command. The ship would not be left without someone at the conn.
Boimler has a letter of recommendation he wants Mariner to sign, printed on paper. Starfleet does not use paper in the 24th century.
The Cerritos comes out of warp directly into a debris field. The ship's sensors should have easily informed them of the danger.
Boimler's new crew mates on the Titan notice that his "PADD is blowing up" with messages from Mariner. PADDs are not communication devices. The only way a PADD would be full of messages is if Boimler had intentionally transferred them from his communications console to the PADD, which would make no sense in general and especially not if he is ignoring the messages, as shown.
Two points regarding ships' registry numbers: the USS Solvang is a new ship. Yet its registry number -- NCC 12101 -- is lower than the Cerritos' -- NCC 75567. And Capt Dayton's previous ship that was destroyed - USS Rubidoux - had a registry number of NCC-12109. Since the Solvang is a newer ship, its number would be expected to be higher, not lower.
However, NCC and NX numbers are building contract numbers. It's entirely possible that the Solvang was commissioned first but construction was delayed, making it a newer ship than the Cerritos or the Rubidoux.
Boimler says that he detects ship components from over 30 different species on the Pakled vessel, but the diagram he displays only shows 9 distinctly identified components.
After the Pakleds' initial attack on the Cerritos, the Pakled captain appears on the Cerritos' viewscreen to taunt them. No one made mention of an incoming transmission and no commanding officer gave the order to accept any such transmission. An incoming transmission shouldn't just appear on the ship's viewscreen like that.
While Freeman and Mariner's relationship may not be common knowledge among the crew, it would still be a matter of public record. Anyone who checked either of their service files would know that Mariner is the captain's daughter.
The bridge crew of the Cerritos has to resort to using hand-to-hand combat and melee weapons to repel the Pakled invaders after being cut off trying to get from the bridge to the armory. It has previously been established that there are weapons stored on the bridge of Starfleet vessels, including the Cerritos. The crew could have armed themselves before fleeing.
When the Titan warps away at the end, the exocomp Peanut Hamper is seen floating in space and calling for help. However, she had beamed into space in the Kalla system. There's no reason why the Titan would've been in the Kalla system; they had escorted the Cerritos back to a star base and taken Boimler on as a new lieutenant junior grade.
Boimler is looking to get "promoted" to the USS Sacramento, thinking it would be a huge career boost. As the name "Sacramento" is the name of a city in California, it is reasonable to presume that the Sacramento is another California-class ship, as all Cali-class ships have been named for cities in that state. California-class ships are all shown to be workhorses of the fleet, lacking in any amount of prestige. There is no reason service aboard the Sacramento should be more of a career boost than service aboard the Cerritos.
Mariner considers applying for a transfer to the Sacramento, as no one would know her there. It has been previously established that the Cerritos was her "last chance" to salvage her career and that if she cannot cut it on the Cerritos, she would be drummed out of the fleet. She would definitely not be able to continue her so-called "Robin Hood" ways on the Sacramento, as her mother's position on the Cerritos is the only thing saving her from already being discharged.
When Boimler is teasing Mariner about being the captain's daughter on Beta III, he closes his eyes and makes kissy noises. It's at this point that captain (overhearing this on an open comm line) beams Boimler and Mariner directly onto the bridge. Boimler is continuing the kissy noises. But he certainly would've stopped after feeling himself being transported.
Lt. Levee gives Mariner a list of ideas on how to decorate the captain's yacht, hoping that she passes it along to her mother, Captain Freeman. The captain's yacht is, as the name suggests, reserved exclusively for use by the ship's captain. Other crew members do not have access to it. Levee would have no reason to be concerned with its decor.
Mariner complains the previous day, Lt. Commander Billups referred to her as "Jenn," asking, "Who here is named Jenn?" There is an Andorian officer named Jennifer on the Cerritos, whom Mariner crossed paths with (and mentioned by name) in "Cupid's Errant Arrow," and with whom Mariner has an ongoing rivalry. Jenn is, of course, a common shortening of the name Jennifer, so Mariner is well aware of at least one potential Jenn on the ship.
Mariner states that Boimler technically outranks her. They are of the same rank, she has more years of service, and he has no positional authority over her. There is no metric by which Boimler outranks Mariner.
Captain Freeman orders that as soon as the engines come back online, Mariner should get them the hell out of there. The port warp nacelle was torn from the ship, so warp speed is out of the question. They would have to proceed at impulse at best. It would take them literally years, if not decades, to make it back to Starfleet without warp speed.