Does anyone know if there exists a test suite (written in QUnit, for example) that tests for ECMAScript 5 support?
The most "official" test suite is probably ECMAScript Test262, which is being developed by members of Ecma TC39.
There's also the open-sourced ECMAScript 5 Conformance Suite developed by Microsoft, and Google's Sputnik, which tests "all aspects" of ECMAScript 3 and parts of ECMAScript 5. Both of those suites are included as part of ECMAScript Test262.
If you just want to test for basic support (e.g. existence of new functions, not whether they behave correctly in every case), there's the ECMAScript 5 compatibility table, which includes a column for the browser you're using.