I played my SL through a genz Benz neo 212t a few times, and even more through an Uber410. It sounded great through the Über, but when the sound guy took a direct out from the amp the foh was really honky compared to my stage sound. I'm now playing through a fEaRful 15/6/1, so what I hear from the cabinet matches the FOH much better. So the setting I mentioned was great through a pretty flat-response cabinet with great low-end extension. At the last gig, very very loud with no PA support, my drummer and audience members commented on how clear and yet powerful each note sounded. With this setup I can finally play on the neck pickup soloed and still hear every note without boominess, all the way down to the low B.
Yes, choice of cabinet makes a huge difference. For me, a PA-style cab like a fEARful/Baer/thunderchild/barefaced kind of takes the cabinet out of the equation (a little), as those kinds of cabinets try to reproduce the signal without too much coloration. I like that, but you may not.