This subject is quite worth revisiting from time to time so, the following is my two pennies worth:
Here in Alaska, The Anchorage Police Department, as well as the Alaska State Troopers / Fish & Wildlife Dept. included, not to mention various smaller Police Departments within Alaska (Fairbanks, Palmer, Juneau, Wasilla and so-forth) have periodically had to shoot grizzly, moose and black bears, not all that uncommonly, over the decades.
Also of interest in regards to the shotgun slug penetration question is the fact that, many of the above agencies have had to shoot through car windshields and car doors, along with various and sundry makeshift barricades, such as furniture piled up and such likes as well.
The end result of these shotgun slug shootings are as follows:
As others have already said here, the old fashioned "Foster" style slugs are too soft and not possessed of the all important sectional density necessary to penetrate well.
This should not be surprising since they were designed primarily for hunting white tail deer at close range.
Others here have mentioned the Brenneke shotgun slugs.
Back in my time, we used the older design Brenneke with the wad fastened to the bottom of the projectile and in 2 & 3/4" at that.
They worked well enough as far as I ever heard tell but, all I personally ever shot with them were targets and they went right through them - LOL.
A chap I formerly worked with (Ed Marsters), bagged a job with the state as a firearms trainer, after he retired from his city job here.
Ed tells me that the best of the best in 12 gauge for penetration these days, is the Brenneke 3" magnum Black Gold slug, live factory ammunition.
Another thing that I found interesting on the shotgun slug subject was that, I test fired a semi-auto 10 gauge shotgun (Ithaca) on an automobile, with slugs and buckshot both.
The buckshot was not very impressive but the 2 ounce slugs / 3.5" magnums, not only shot through the car doors with ease but, they also punched very perfect looking round holes through the wheels, steel rims, brake drums and all !
Evidently the two oz. mass, even though of the otherwise feeble Foster design, had enough momentum to behave entirely different than the same 1 oz. 12 gauge version.
Well, my fingers are getting tired so, it must be time for beer.
Out.