Single cut basses have just one cut out, compared to the more usual double cut away model. If you can picture a classical guitar in your head, or an old steel-string acoustic, you’ll know what a guitar with no cut aways looks like. The guitar body near the neck on the top string half was originally cut away to provide better access to the high frets. I’m not sure when or why in the life of the guitar the second cut away became common, Leo Fender just made them that way, giving them their two distinctive ‘horns’.
Bass guitars inherited the double cutaway design of their higher pitched brethren, via Leo Fender, even though the double bass has no cut aways. However, with through-neck designed basses, there are a number of reasons to go for a single cut design.
The single cut allows for more contact between the neck and the body, adding stability. It also allows for more wood in the body, which affects the appearance, and potentially the sound. Their looks are slightly controversial - many players think they look ugly – and some argue that the design restricts access to the upper frets, but it never has for me. Makers like Veillette Guitars do make bolt on models (a nice diagram from them here).
Back to Aquilina and their Shelby bass for a moment, this imposing looking 7 string shows how the neck looks from behind, quite different as you can see:
While I’m here it would be just wrong not to show you the front of this lovely bass too:
The design also shifts the balance of the bass, increases the weight and enhances the sustain. You may see those as plusses or minuses. I think the choice to go for a single cut design is one of personal preference, since there aren’t dramatic differences in tone. They sure do look different though, don’t they!