I going to be moving into an apartment next month and I probably won't be able to use an actual amp (especially at night which is when I usually like to practice).
Are there any good amp simulators that I can use to practice through headphones?
Can anyone recommend a good one and a way to set it up so it sounds decent?
Line 6 POD HD series, Eleven Rack, Positive Grid (if you have an iPad or other iThing), or if you want to throw thousands of dollars at top grade studio equipment you have the Kemper and AxeFX.
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Amp simulators have become just as good as any analog amp today. I saw Accept live last week and they were using Kempers, as they also did recording their latest album I believe. They're just one of many who have switched over, and many more will probably follow.

The problem though, is cost. Kemper is expensive.

Among the affordable I have experience with a Line UX1 interface together with Pod Farm, and Guitar Rig. I think Guitar Rig is superior to Pod Farm, which sounds good but very different from a real amp. Both solutions are cheap and I don't think you can go wrong with any of them for practicing with headphones, to be honest.
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You can do this two ways; 1) Software (Amplitube, GuitarRig, ReValver, etc), or 2) Hardware: Multitude of MultiEFX boards with Amp/Effects sims from Boss ME25, 50, 70, 80, GT10, GT100, Line 6 has desktop POD/Rack format POD and floor POD HD300, 400, 500, there are others from Xoom and the list goes on and on.

If you go for a hardware solution use is as simple as plugging in your guitar, headphones, dialing up a preset or creating your own and playing. Depending on your level of familiarity and confidence with amp tweaking and effects settings, virtually every hardware unit can provide tones that will satisfy if not delight you. Some are better at amps some are better at effects but if you go with Boss or Line6 you will have thousands of sounds available between what comes on the unit and the online user patch exchanges and manufacturer updates.

Also, if you want to amplify your tones (no headphones) w/o pissing off neighbors a decent sent of monitors is a great addition. This would apply for a software solution as well, though you will need an interface to connect your guitar to PC/Laptop/Tablet...SEE SOFTWARE

If you go Software I personally recommend Amplitube 3, the free version is pretty robust and comes with a number of amps, effects and rack effects. The primary consideration here is that you will need an interface to connect your guitar to your mac/pc/tablet...again, sound quality will depend on 1) Interface you buy, 2) Tone tweaking ability.

Hope this helps, I do both extensively as well as full analog pedal board/amp setup at home so feel free to ask any questions and I'll be glad to further assist.
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I like the Line 6 HD500 "pedal" and UX1 and 2 interfaces I have, though another guitarist I know didn't like the HD500. Difference in taste and/or ability to hear stuff I suppose.
I recently discovered that the all-analog Sansamp Character pedal is very good as a headphone amp, but the output is only mono (one ear, which I like), and each pedal is specific to one make. I have the "British", a Marshall emulator.

I also liked the Pandora Px4 before it stopped working.
I use the Line 6 Pods on a regular basis, both for home headphone practice and for gigging. For the latter, I just run them through a power amp and a full-range speaker system (something like a PA speaker).

I have several of the little Korg Pandoras -- a PX3D, a PX5D, and one of the new Minis (in a box, a gift. Haven't pulled it out yet). These are battery powered (though there are AC adapters available) and, except for the Mini (which is tiny) about the size of a pack of cigarettes. Enormous capabilities in those things, including amp/FX/cabinet sims, tuners, metronome, slower-downers, pitch shifter, bass and drum tracks and even a phrase trainer (loops a portion of a song over and over so that you can practice with it) and much much more.
The Line 6 POD HD is what I've been using for a while and after the latest firmware update the amp simulations are pretty good. It's a robust reliable unit and, for the money compared to the Axe-Fx it's a great buy. Works well will headphones but if I wanted to listen through a speaker system I put it through an interface and through a pair of monitor speakers which is good for recording and helps fine tune the settings.

Set up wise - The amps sims for the Mesa Triple Rec (Treadplate), the Bogner Uberschall are my favourite for metal and rock. You'll need to use a noise gate and overdrive with the sims to help tighten/boost the mids and control any unwanted buzz.
I've been using Pod Farm through the Pod Studio UX1 for a few months now. Didn't like it at first but after a few months experience with it I've found a lot of sounds I really love. Thinking about upgrading to the platinum version now.