When the three prisoners are beamed into space, the viewer sees them instantly freeze solid. This is a common misconception and mistake in science fiction. Since there is no air in space, the is no convection for such quick heat loss. A person would die of suffocation long before they would freeze.
The characters repeatedly mispronounce the name "Kahless," the Klingon deity, with a "sh" sound at the end of the word. Decades of prior Star Trek material established it as being pronounced with an "s" sound.
When Burnham and Tyler are beamed to Harlak, in the first wide shot there are significantly more footprints on the ground than they could have made since their arrival and with the way they moved - which is barely at all. (This of course assumes that no one else--the rebels that ambush them, for instance--has made footprints in the recent past.)
When the three prisoners are beamed off the ISS Shenzhou into space, then dematerialize standing straight upright but rematerialize in space with their limbs spread.
At 19:18 when Lt. Cmdr. Stamets (Anthony Rapp) is in the "Spore Drive Chamber", a 20th Century handicapped commercial grade automatic door opener/closer can clearly be seen in the upper left corner of the screen attached to the chambers entry portal.
At 45:00, Lorca's accent deviates for about 15 seconds into a British accent instead of the character's standard American accent.
Tilly mispronounces the word "temporal."