The second time Booth sneaks to first class, Bones earphones are off when the camera is aimed at her, but when it's aimed at Booth, her earphones are still on. It goes back to her and they're down again.
Before discovering the body, Booth and Brennan run down a staircase in the airplane, implying that they're on a double-decker aircraft such as a 747 or A380. However when the camera cuts to a shot of the plane from the outside, it's visibly a single story aircraft.
The first time Booth sneaks into first class, he catches the 16 year old boy stealing alcohol from an unattended beverage cart that is right next to him. The flight crew would never leave a beverage cart unattended like that for exactly that reason. They would be responsible for anyone underage who stole alcohol from the cart.
Bones seeks out the passenger with trigger finger because he might have yellow shooting glasses with him. Yellow shooting glasses are a real thing, but "Trigger finger" is named because of what the finger looks like with this condition and has absolutely nothing with having fired a gun.
The 16 year old boy when finally accused and waiting for the return flight, has one end of the handcuffs on him but the other end appears to be just dangling. However, the other end is fastened to a seat belt.
A Great Circle route to Shanghai from Washington does not fly over Greenland. It misses by about 500 miles to the West.
Booth does not actually finish the Miranda Rights. It is required for the arresting officer to say them completely otherwise the arrest can be thrown out.