I love watching the show and spotting things that take you out of the reality of the show and reveal how things are being done form a production level.
For example sometimes it is obvious that they are redressing sets. This happens quite a bit between federation sets but you'd be surprised how often they do it. A good example is the Discovery hallways and Transporter rooms have snap on/off panels so they could switch between it and the Shenzhou. A sad example is when they converted the Discovery transporter room into the the Bahoul pool room is VERY obvious.
You can see plenty of things that look suspiciously like iPads (which they are).
I was watching Season 1 and noticed when Saru and Michael were walking down the hall eating blueberries you can see a gap in the wall which shows the plywood that the set is made out of

You can also see they cut had the actors walk back a little and pretend the hall is longer as they walk by the gap twice.
You guys notice anything else?