When Shaxs enters the turbolift that Mariner is lubing, he turns around just as the doors close to reveal that he's wearing 3 solid rank pips, indicating he's a commander. But in all other scenes, he's wearing 2 solid pips of a lieutenant. Since the credits list him as Lt. Shaxs, it would seem the commander pips are erroneous.
When Tendi accidentally messes up O'Connor's sand mandala, she tries to fix it by asking the replicator for some more colorful sand. But there is already sand in the replicator as she requests it.
When Tendi and O'Connor are arguing in the flooded engineering compartment, an explosive spore detonates between them, throwing them away from each other. When they are then seen sinking in the water, they are closer together than when they were before the spore exploded.
When Tendi and O'Connor are arguing in the flooding engineering compartment, they are standing in stomach-high water. A sport detonates between them and the water is suddenly deep enough that they sink a good distance and Tendi has to grab O'Connor and swim them both to safety.
While digging with handheld rocks, Captain Freeman suggests that Mariner use a shale rock, as it might be better for the task. Shale is a sedimentary rock primarily made of mud and clay and splits very easily. It would be very ill-suited for use as a digging tool.
When the senior officers are playing poker, the fold/bet/raise process is done in the incorrect order. Starting with Freeman, the order should've gone clockwise: Freeman, Mariner, T'Ana, Ransom, and Shaxs. Instead, it goes Freeman, Ransom, T'Ana, Mariner.
When Mariner is promoted, she is given a second, solid gold pip to indicate her rank. Two solid gold pips indicate a full lieutenant. An entire rank, that of lieutenant junior grade (one gold pip and one black pip) has been skipped over. An officer would not be given what is essentially a double-promotion.
The design of a generation ship is to have conscious crew and passengers, passing control from one generation to the next. Therefore the crew would not be mummified in in cryo-chambers, they would either be present or dead. However, since "generation ships" are entirely fictional at this point in time (early 21st-Century Earth), this can't be considered a FACTUAL error.
When Freeman and Marine break through a rock wall into a turbolift shaft, there are already vines leading from the corridor they crawled through down the shaft for them to climb down on. There had been a solid rock wall there just moments prior.
When the Merced's crew is beamed over to the derelict generation ship, only about a dozen or so crewmen materialize. A California-class ship like the Merced has a standard crew compliment of around 300.
When Mariner is forced to go to the Officer's Lounge for management training (which ends up being essentially a talent show for the senior staff), Ensign Barnes is seen in the crowd. Ensign Barnes is not a ranking officer.
Captain Freeman and Ransom conspire to make Mariner so unhappy serving on the Cerritos that she will request a transfer of her own volition, rather than Freeman having to reassign her. The premier established that the Cerritos was Mariner's "last chance" at a Starfleet career and that if she couldn't hack it there, she'd be drummed out of the service. A transfer to another ship should not be a viable option.
When Mariner is promoted, she is given a new uniform in a gold color. Uniform colors indicate division of service, not rank. The gold uniform is for operations and engineering, while the red she typically wears is for command and bridge crew. There is no reason her promotion would come with a transfer of departments, especially as she has no established background in operations or engineering.
Tendi saves herself and O'Connor from the flooding engineering compartment by wedging an exploding spore into the wall, which blows a hole in it and allows the water to empty into the corridor. If emptying the water into the corridor was an option, that could have been accomplished by someone simply opening the door.