I started playing pedal steel guitar a few years back. Steel players generally like REALLY clean tone. For home practice I use my standard Zum Steel Encore to Hilton VP and then to pedalboard. From their it's either into my Milkman Half and Half (a 300 watt Tube pre/ClassD hybrid with 15' speaker) turned waaaaay down or into headphones via my Zoom B1on. It depends on how early in the morning I'm practicing.
6 string with option for overdrive/dirt/etc is a different matter. When messing about with 6 string (Agile LP clone or Squire strat), or when having guitarist friends over to jam in my small music room, I simply plug the guitar into an old POD 2.0 which I run either into headphones or my Sunn Alpha 4 PA. For 20 year old tech, it has some impressive sounds, breaks up quite realistically and with so many knobs is easier to use than I expected. I picked up the POD 2.0 because it was only $50 bucks and I had already been given the original large floorboard which doesn't work with any later models.
If I wanted a budget home practice setup starting from scratch, I'd probably skip the 2.0 (hassle to interface with PC as it doesn't have a USB hookup and has no stereo audio input) and pick up whatever Pod XT, or PodX3 I could find locally for under a hundred bucks or maybe a Pocket Pod. Any of those would be an excellent headphone amp, a good recording interface (though I don't think the Pocket Pod is), a useful platform to put in front of whatever amp you want to play softly, and if you're so inclined, there's a whole world of tweaking you can do once you hook it to your computer.
There are alot of other devices that do similar things, but suffice to say I really think digital is the way to go if you really want quality amp breakup sound at very low volumes. Save your pennies for a good stage amp and let cheap digital devices do the work for you at home practice.