Would self grounding outlets (15/20 amp) be useable in metal boxes fed by BX? The house was built in 1959 and all original wiring is BX! Any new wiring (Romex/style) has the appropriate ground wire!
Properly installed BX cable armor does provide a path back to the main panel. While in older BX this path is no longer Code-suitable as an equipment grounding conductor, the installation of self-grounding type receptacles would not actively harm anything, and would be at least somewhat useful in case of a hard fault.
However, a non-self-grounding type receptacle will not provide a satisfactory ground path, as it depends critically on the tightness of the fasteners holding the yoke to the box in order for the yoke to be grounded in that case. The spring-loaded clips in a self-grounding receptacle are designed to maintain this ground path, even if the fasteners work loose.
If the BX is properly installed, the cable's metal jacket should provide a ground path. BX should always be connected into metal boxes with the appropriate clamps so that the box itself is grounded. You can test for this using a voltage tester - the box should test at 120V to hot and 0V to neutral.
If that's the case, a self-grounding outlet receptacle installed in the box will be grounded. The ground terminal on the receptacle is connected to the box mounting hardware, so you're just having the box provide the ground path instead, and connecting it via the existing mounting hardware. Note that this method of grounding (relying on the BX jacket) is no longer code compliant, but it is functional and safe as long as there is a reliable grounding path.
If the jacket is not properly grounded, you can use a GFI to provide some ground fault protection. You can also run a new ground conductor in some cases.