At the staff meeting, Geordi says it will take the force of at least 5 photon torpedoes to dissipate the wave. When the Enterprise intercepts the wave, they only fire 3 torpedoes, yet somehow the wave is still effectively dissipated.
The computer twice reports the location of Alexander. But Alexander is not wearing a comm badge, which is how the computer keeps track of where anyone is. However, that is the primary method. The computer can use internal sensors to track life-signs. Distinguishing different adult humans is presumably more difficult, which is why comm badge signals are used. On the other hand, finding the one Klingon child on board would be fairly easy.
After the test ship riding the Soliton wave is destroyed, the wave begins to significantly increase its velocity (thus its energy) and size. However, how the wave has increased in energy is never explained. Correction: the Soliton wave is a (fictional) advanced technology/phenomenon beyond the understanding of current science. Just because it's not explained onscreen how it increases in energy as it travels does not mean there is a plot hole.
In the holodeck scene, Worf orders the computer to "freeze program". Yet the smoke behind him as well as the ground fog around Alexander still continue to move.
When Alexander has the field trip and stands next to the Gilbos terrarium, his head and shoulders can be seen on level with the creature. In the next scene he is barely able to look in the cage at all.
If the speed of the wave increased unexpectedly, it should have missed the target planet due to planetary motions.
In order to position the Enterprise ahead of the soliton wave to dissipate it using photon torpedoes, the crew state that the ship will need to pass THROUGH the wave. This subjects the ship to needless danger, causing injuries and severe damage. This is another of many times that the writers seem to forget that space allows travel in any direction; the Enterprise could have safely flown over or under the wave and then moved into position ahead of the wave.
When the Enterprise is chasing the Soliton wave, Picard is informed it is traveling at warp 7.2. He instructs the helm officer to increase the ship's speed to warp 7.21 and prepare to go to warp 7.3. It makes no sense to only give them a minute speed advantage over the phenomenon they're chasing. Increasing to warp 8 or 9 would allow them to overtake it much faster and give them precious extra minutes to enact their plan.
Kahless is pronounced here as "Kah-less" while all other Trek productions about this character call him "Kay-less".
When the senior officers enter the bridge after the initial briefing on the Soliton Wave by Dr Ja'Dar, Deanna, speaking to Worf about Alexander's teacher, should have referred to MISS Kyle rather than MRS Kyle.
Upon entering the inflamed biolab, Worf and Riker begin shouting for Alexander. When he does not respond, they continue shouting and begin visually searching the smoke-and-flame filled room. They were aware Alexander might be injured or unconscious, and time is of the essence, yet neither thinks to use a tricorder to scan for him. Even in such a chaotic situation, surely two seasoned officers would keep their wits about them enough to use the most appropriate tools possible.