I know how DSC works on other BMW vehicles, I presume the E60 is the same.
Default is DSC on. No lights on the dash (well, unless the system is correctly engaging to overcome too much throttle application). DSC includes dynamic stability control, dynamic traction control, etc. In simple terms, stability control addresses sliding sideways while traction control addresses wheelspin. Not sure why you would want to turn it off, unless you are on the race track. Statistics have shown that it reduces accidents. If you want to play around with it, which is a very good idea in terms of learning what the car can do, get on an empty snow-covered road or parking lot, and toggle it on and off. There is an amazing amount of stability added by the system, IMO.
You can turn DSC off with the pushbutton, and then DTC shows on the dash. What that means is that while you have disabled the stability control features, you still have the same traction control. Same comments as above, not sure why you would turn it off unless you are on the track.
You can then hold the DSC button and that then turns DTC off as well. In my vehicle I get a bright caution light, basically the car's way of asking me if I am sure I know what I am doing. However, this is the setting that has some value. If you are stopped on a snow-covered road, such as a steep driveway, and you can't move because the vehicle has very limited traction and the DTC is automatically reducing throttle to limit wheel slip, disabling DTC lets you spin the wheels and effectively churn your way out (unless you are really stuck....).
I always leave DSC on, especially on snowy roads, but if you are stuck in a snowbank and can't accelerate the engine because the DTC system is disagreeing with your right foot, then spinning the wheels can help. As soon as you are moving again, turn DSC back on. Just don't overdo the wheel spinning, as there are a limited number of situations where you will get moving when DTC doesn't think you can do so.
Hope that helps. If I have applied E46 and X5 logic to the E60 inappropriately, somebody please chime in.
2008 535i 6 MT, Premium, M Sport, Space Grey
2007 X3 3.0si 6 MT, Premium, White
2003 X5 3.0 auto, Premium, Titanium Silver
2004 Z4 3.0 SMG, Sport, Premium, Maldives Blue
2002 325xi 5 MT, Steel Grey