u did not provide a sample image of a tucked shirt person LOLShould casual checkered shirts be tucked in? (refer to the images below)
Mostly this. Presuming it fits (the example pictures are not good examples of this), you can wear it either way. The dressier the pants though, the more necessary it becomes to tuck. Likewise with the checkers, the bolder the print the less formal it becomes.Not sure if i've ever seen someone tuck in a checkered shirt, but can't say i've been looking haha.
As for shirts in general: pants with a dress shirt or business shirt, definitely. Jeans with a casual shirt, depends. In the first picture that guys shirt looks waaaay too long in the back, I also think the curved hems on the bottom are meant for tucking in but I've seen people wearing them untucked before - just not anywhere near that long.
Whatever you do, if you tuck in the shirt make sure you are wearing a belt.