How can I check that the nginx gzip_static module is working?

Use strace. First, you need to detect PID of nginx process:

# ps ax | grep nginx
25043 ?        Ss     0:00 nginx: master process /usr/sbin/nginx -c /etc/nginx/nginx.conf
25044 ?        S      0:02 nginx: worker process

Ok, so 25044 is the worker process. Now, we trace it:

# strace -p 25044 2>&1 | grep gz
open("/var/www/css/ymax.css.gz", O_RDONLY|O_NONBLOCK) = 438
open("/var/www/css/patches/patch_my_layout.css.gz", O_RDONLY|O_NONBLOCK) = -1 ENOENT (No such file or directory)
open("/var/www/yaml/core/iehacks.css.gz", O_RDONLY|O_NONBLOCK) = -1 ENOENT (No such file or directory)
open("/var/www/js/koznazna5.js.gz", O_RDONLY|O_NONBLOCK) = -1 ENOENT (No such file or directory)
open("/var/www/css/ymax.css.gz", O_RDONLY|O_NONBLOCK) = 216

As you can see, it is trying to find .gz versions of files.

Change the content of the non-gzipped file. And then touch both files (simultaneously—that is: in the same instantiation of touch). If when you load the file in an browser (cache-wiped) you get the non-changed file, then nginx served the static-cached-gzipped file.

An easy way to avoid “did I just fetch the cache?” worries is to fetch from the command-line with curl since curl doesn’t cache.

I would disable automatic compression and log gzip_ratio:

http {
   gzip off;
   gzip_static on;

   log_format log_with_ratio "... $gzip_ratio ...";

   server {
      access_log /var/log/nginx/file.log log_with_ratio;
   }
}

Note, that you may override gzip, gzip_static and access_log per server and location levels.