Found it in QV11 SR2, didn't test in the other versions.
Drop-down select indicator (down-arrow) doesn’t show up in tables in AJAX client and web view, while worked in IE Plugin. The reason – it depends on the table properties, selection indicator. If turn off selection indicator in a table, there is no drop-down select in web view.
Fix is easy - turn on selection indicator - but apparently it is a bug. I don’t see any logical reason why it should behave this way in web view.