count all occurrences of string in lots of files with grep

grep -c string *

...
file1:1
file2:0
file3:0
...

grep -c string * | grep -v :0

...
file4:5
file5:1
file6:2
...

cat * | grep -c string

grep -o string * | wc -l

cat * | grep -c string

grep string * | wc -l