My outer basement walls...
I just finished insulating my basement. As mentioned there is a lot of way to do this and some of those ways depend on the application/context. Spray foam is the best way to go but it has downsides; it's expensive and you have to fire rate any exposed foam (ie. drywall or spray fire retardant). Upside is no vapor barrier needed and you don't have to worry about moisture build up.
Besides spray foam, no other method will perfectly seal vapor or stop moisture build-up. When you insulate using other methods, there are four things that are a must: 1) moisture barrier, 2) insulation, 3) vapor barrier, and 4) breathing/venting. Most handle the first three but forget the fourth. Moisture/vapor WILL build up *behind* the vapor barrier because no seal is perfect and if moisture/vapor has no where to go, mold/water damage will result. In my basement I left the concrete bare at grade and above; i.e. no moisture barrier, tar paper, or glued on rigid foam above grade. Why? Because the bare concrete breaths above grade to the outside. I got this tidbit of info from the NRC. Of course I still put in insulation (not touching the concrete) all the way into between the floor joists, then VB tuck taped everywhere. Hope this helps, you can read my thread for info or for pics.