The problem with LSD's on ice
LSD's are great where one drive wheel has good traction and the other doesn't (e.g., left wheel on dry pavement and right wheel on wet pavement) because both wheels still get power even though one can't use it at the moment. And they are even better when powering through a tight corner where body lean unweights the inside tire but the outside wheel keeps driving (an open differential would allow the inside wheel to spin and reduce the outside wheel to coasting).
When both wheels have no traction (such as an icy road or one covered with packed snow), though, I'd rather NOT have a LSD. The reason is that when both drive wheels are spinning as they might when your engine overpowers the available traction, other torque forces in the car start to affect the direction of your hockey puck. What usually happens is that the tail end (in a RWD car) will head toward the curb putting you sideways to the flow of traffic.
An open (standard) rear end won't do that in most cases; instead, you will just spin one wheel, the other one won't turn at all, so you stay stuck where you are.
Why do I prefer no movement? Because then the idiot behind you that can't stop will hit you in the energy absorbing bumper and drive your head into the headrest instead of hitting you in the driver's door, driving your head into the side window glass!