find ./ -type f -size 0

or

find ./ -type f -empty

This commands will find all zero size files in the current directory with sub-directories and then print the full pathname for each file to the screen.

• ./ means start searching from the current directory. If you want to find files from another directory then replace the ./ with the path to needed directory. For example, to search everything under the system log directory you need to replace ./ with /var/log.
• -type f flag specifies to find only files.
• -size 0 and -empty flags specifies to find zero length files.

To find and then delete all zero size files, there are variants you can use:

find ./ -type f -size 0 -exec rm -f {} \;
 
find ./ -type f -size 0 | xargs rm -f
 
find ./ -type f -size 0 -delete

• The xargs will cause all the filenames to be sent as arguments to the rm -f commands. This will save processes that are forked every time -exec rm -f is run. But it fails with spaces etc in file names.