They do, though it's not that wide and some Bigsby screws would probably miss it. Unless you have a Bigsby designed for a solid body, most Bigsbys will extend back to the strap button, which is their main securing place. For those that have screws for the top (several don't at all), those screws are mainly there to stop the Bigsby moving sideways, so they don't need a huge amount of strength in an upwards direction.