Ignoring specific errors in a shell script

In order to cause bash to ignore errors for specific commands you can say:

some-arbitrary-command || true

This would make the script continue. For example, if you have the following script:

$ cat foo
set -e
echo 1
some-arbitrary-command || true
echo 2

Executing it would return:

$ bash foo
1
z: line 3: some-arbitrary-command: command not found
2

In the absence of || true in the command line, it’d have produced:

$ bash foo
1
z: line 3: some-arbitrary-command: command not found

Quote from the manual:

The shell does not exit if the command that fails is part of the
command list immediately following a while or until keyword, part of
the test in an if statement, part of any command executed in a && or
|| list except the command following the final && or ||, any command
in a pipeline but the last, or if the command’s return status is being
inverted with !. A trap on ERR, if set, is executed before the shell
exits.

EDIT: In order to change the behaviour such that in the execution should continue only if executing some-arbitrary-command returned file not found as part of the error, you can say:

[[ $(some-arbitrary-command 2>&1) =~ "file not found" ]]

As an example, execute the following (no file named MissingFile.txt exists):

$ cat foo 
#!/bin/bash
set -u
set -e
foo() {
  rm MissingFile.txt
}
echo 1
[[ $(foo 2>&1) =~ "No such file" ]]
echo 2
$(foo)
echo 3

This produces the following output:

$ bash foo 
1
2
rm: cannot remove `MissingFile.txt': No such file or directory

Note that echo 2 was executed but echo 3 wasn’t.