You can do something like:
Test for execution:
if [ -x "$file" ]; then #do_somethingfi
Test for writing:
if [ -w "$file" ]; then #do_somethingfi
Test for reading:
if [ -r "$file" ]; then #do_somethingfi
stat FILENAME --print=%A
see man stat for more options - you can check the group/owner/etc.
man stat
You could use test
The options -r -x -w test for read,execute and write permission.In a shell script the test command is usually entered as [