Check visually that none of its pins have become wrapped under its body instead of being inserted into the board. If the output is within a volt or so of either power rail then either it has failed, or there is an excessive voltage at its inputs. Measure the voltage between the inverting and non-inverting inputs, it should be within millivolts of zero. A few millivolts of offset at the input of a system with a high DC gain can be amplified to the point that it saturates the final stage. Many op-amps have offset adjust facilities which can reduce the offset by something like a factor of 10.
Offsets are temperature dependent.
Instability Instability typically appears as high-frequency 'fuzz' on the output signal oscilloscope trace. Breadboard circuits are very prone to it because of inter-track capacitance and long components leads. Don't use op-amps with an unnecessarily good frequency response, e.g. use a LM741 instead of an LF411 where possible. Check the signal at the input and output of each section, using an oscilloscope if appropriate. It is possible for a faulty section to load its predecessor in the chain.