I see quite a few web applications which have a very similar layout, so I assume there is a particular framework they’re using in common. The top bar has this sort of layout:


And there’s a side-bar, which is often a bit Apple-esque:

Does anyone know if this is some kind of common (hopefully open-source) UI framework?
The outward appearance of a web application is never an indication of which framework was used to build it. The things you’re showing are all the results of styling and can be done with any web framework.
You can, however, look at the source to see what kind of javascript references there are and get an idea that way. Certain frameworks will sometimes give clues (asp.net will have _viewstate, etc.).
For example, in looking at the source for Trello you cain find a js reference that has a comment header section that gives you lots of clues: