Picard and Riker meet the maintenance crew in an airlock on Deck 25 (seen on the panel on the airlock door), but, during their walking conversation, they turn a corner and are suddenly in Main Engineering (established in the series as being on Deck 36).
When Picard and Riker learn about the red alert, Minuet's hands are shown in a close-up as being separate and curled. When she is shown in the background, her hands seem to be together as if clapping. This effect is created due to the camera angle and the distant, slightly blurry picture. With a closer look one can see that Minuet holds in fact her hands in the same position throughout the entire scene. However, she has her head turned away in one shot, while in all others, she looks toward Picard and Riker.
When the Enterprise is being evacuated, there is a scene with people waiting in line to use the transporter to get off the ship. The next scene shows a group of people (including Wesley Crusher) being beamed off the ship; however, only five people beam off. The Enterprise transporter pad has a total of six circular "small" pads and one larger center pad. In theory, about 7-8 humanoid people could beam off the pad at once, yet during this "emergency" evacuation, the transporters are not being filled to capacity (as they should be in a true emergency situation).
At around 32 minutes, when Picard and Riker activate the self-destruct countdown, the timer counts down significantly more slowly than actual time, even during the time in which the timer display is shown to be counting.
It would be impossible to evacuate the entire complement of the Enterprise and then clear the space station to minimum safe distance in but 4 minutes.
When Riker asks the computer why the power has shut down, breaths can be heard in between the computer's spoken lines. A computer would not need to breathe.
When the Enterprise enters starbase 74, the image has been flipped: the ship's registration number is reversed on underside of saucer section.
While Riker is in the Holodeck dancing with Minuet, you can clearly see the band is not playing the music on the soundtrack, as their timing is off.
None of the musicians (including Riker) are actually playing their instruments. They are all faking, badly. For example, the drummer isn't actually touching the snare and cymbal with his brushes. Riker's slide movements do not match the music he is supposedly playing (in spite of the fact the Jonathan Frakes can actually play the trombone).
When Riker and Picard first want to leave the holodeck, Minuet tries to delay them so they will not uncover the Bynars' activities. The pair ignore her and leave anyway. The Bynars were able to cut off communication from the rest of the ship to the holodeck and vice versa, making Picard and Riker unaware of the red alert and evacuation order. They were also able to alter the ship's sensors so that Data would think Picard and Riker had already evacuated, as well as alter the anti-matter containment field to create the alert status in the first place. If they were able to accomplish all this, surely they could have locked the holodeck doors.
When Data and La Forge determine that the antimatter containment is failing, Data orders Wesley Crusher (on the bridge) to set a course for the ship away from inhabited planets once the Enterprise is evacuated. However, when Data and La Forge are later on the bridge just before leaving, Data is setting the course. Did Crusher simply ignore his orders and flee the ship? (Possibly, since he's shown beaming off the ship in the evacuation montage.)
Despite the dire emergency, no-one seems to think to use the starbase's transporters, or any of the shuttlecraft to escape from the Enterprise, or to chase it, or to beam onto it as it leaves. And while it's conceivable that a Starfleet starbase only owns 3 non-functional ships, there would have to be several functional ships (if privately-owned) at the base to carry supplies & products to & from the base. Any of them might have been used to chase the Enterprise.
Converting the Bynars' names to decimal; 0 was paired with 3, and 1 was paired with 2. That exhausts the supply of 2-digit binary names (or numbers), which suggests either that there are only 4 Bynars in existence, or that their names are reused so extensively that names would be useless, or that these 4 have abnormally-short names among the total population.
The Enterprise had already undergone a computer upgrade shortly after the events in Datalore (1988). It is very unlikely that another up-grade would be performed so soon. In addition, Enterprise has been operational less than a year. As the ship is designed for extended missions lasting several years, it is never explained why extensive routine maintenance is needed so soon.
It was a mistake for the Bynars to have sealed off the Bridge from turbolift access. They knew that--in fact, their plan depended on--Riker getting to the Bridge to help them.
Throughout the emergency on the Enterprise, Data acts with agitation, excitement, and panic. As he leaves the bridge, Data also breathes a deep sigh and shows worry on his face. None of these traits are characteristic for Data and he should be behaving with a much calmer demeanor.
Picard and Riker dramatically arm themselves with phasers from the Armory before beaming onto the Bridge and finding the Binars incapacitated. They deduce that Minuet may have some answers to their questions, but, when they arrive on the Holodeck to question Minuet, they are unarmed. They would not have wasted time to check in their phasers before questioning her.
Minuet claims to have heard Captain Picard's name and accessed the foreign language databank to converse with him in French. First, because someone has a foreign-sounding name, it doesn't mean they're from that region. Secondly, there's no need for her to access any databank because everyone benefits from the universal translators to understand everyone else regardless of language.
While aboard the starbase, Commander Quinteros is paged stating that the caller is aboard the starbase. They are already on the starbase. There wouldn't have had any need to state where they were.