How to check that cache is working on current page? Some modules silently turn off caching on certain pages. How is the easiest way to check that this is the case and also identify problematic module?
The easiest way is probably to inspect the HTTP response headers. 

For example, with your browser DOM inspector tool ( e.g Chrome's "Network" tab )

Cache Miss

Here's some sample response headers from drupal.org showing a cache miss. In this case, using Varnish however core Drupal cache also sets similar headers.



Cache Hit

Here's one showing a standard Drupal cache hit: