What happens when an op amp misses one of the power rails?
It depends a lot on the topology and the specific devices involved, so without a schematic to review this will be a fairly general answer, but generally the damage conditions are similar to those that occur when a device's IO voltage ratings are exceeded.  Essentially, the op-amps power rails will be floating, but likely pulled towards ground (or the opposite supply rail) by all of the devices connected across those same power rails.  This includes other ICs, as well as voltage dividers, indicator LEDs, bulk/bypass caps, etc.  

So as a a starting condition we can assume that the floating rail--which we'll assume is the negative one since that's the case in your situation--is at the same voltage as the positive rail.  Thus, any voltage applied to an input or output pin that is less than the positive rail is effectively outside of the op-amps supply rails.  On many devices, this will cause current to flow through parasitic junctions or ESD protection diodes from the input pin to the negative supply rail.  If the source impedance is low enough, enough current can flow that the floating negative rail can start to be pulled further negative--essentially the negative rail is being supplied through that signal pin.  Because these parasitic junctions were never intended to supply current to the entire IC, let alone all of the surrounding circuitry, this can cause them to fail.  Likewise, the ESD diodes, if present, were meant to conduct brief surges where, even though the instantaneous current may be substantial, the total energy is limited.  Sustained DC current can cause these to fail as well.  

It's quite common for semiconductors to fail short, in which case, depending on the exact nature of the internal failure, a device may wind up with an input or output pin shorted to a supply rail, or may even wind up with both supply rails internally shorted together, which will obviously disrupt the entire surrounding circuit.  More subtle failures may also occur, that simply compromise the internal function of the device.  Total failures are common, but partial failures, including changes to device parameters, are certainly possible.