Allison tells Carter that all canceled projects are discarded (recycled). However, in a previous episode, the GD storage area is featured and it is stated that all old project material and devices are stored there.
The Phase vest was used in another episode for an entirely different purpose.
When Jack and Lexi are talking in the jeep a white vehicle can be seen in the passenger mirror traveling behind them. However, when they pull over to inspect the leaking pipe no vehicle is behind them and no vehicle continues past them.
Since the vest allows the wearer to pass through matter, both Captain Eureka and Jack should have fallen through the floor the moment they turned it on.
When Zoe, Zane, Pilar and Lucas are trapped underground, they say they have an hour or two before they suffocate, but, after Jack goes through the first door , the camera pans past the exhaust fan in the wall, and it is visible that the dust is being blown into the room from the shaft. So, they couldn't suffocate with fresh air blowing in. And they all should have noticed.
Carter and his Sister are driving to a propellant pumping station, before reaching it, seconds before arriving, when the camera is on to Lexi there is a White Van in the mirror, when they pull over it's gone.
Possibly on purpose. Location given for the underground building connected to the Tesla School is 123N 39W. Since there is no number greater then 90 in N/S this my be in intentional error. Although reversing gives you 39N 123W, an empty mountainous region in Northern California.
At 13:07, an exit sign is clearly visible down the hallway, right before Zoe asks which way they should go to leave the facility. They seemingly don't notice it, despite the camera panning right over it.