Bulbs should be firm and never mushy or rotten—small amounts of surface mold should be okay. If only a small part of the bulb seems to be mushy, try slicing off that part with a knife and stick it in the ground anyway. It’s better to give it a fighting chance than to just assume it’s dead and throw it away. Even chancy plants can live a healthy life if you’re diligent.
Bulbs that have already sprouted will be vulnerable during the winter, so make sure your bulbs have as little new growth as possible. Larger flower bulbs, like tulips, produce bigger plants and tend to come up later in the season than plants from smaller bulbs. Plants with smaller bulbs, like crocus, come up much earlier.