If you notice that Pop Shop is crashing when you are trying to install an application check your free space! There is a very real chance that you are running low on space. The OS will not notify you it will just crash the app.
A colleague of mine was running into this problem. His home directory was on a separate partition and there was not enough free space.
You can run Pop Shop in the terminal by running: io.elementary.appcenter
By running it in the terminal you should be able to see any output if there are errors when you are trying to perform certain tasks.