Can you both supercool and superheat a substance simultaneously? There are certain temperatures and pressures for substances at which they exist as a liquid. Also there are temperatures and pressures for which they usually are at equilibrium between solid and gas. Superheating is where you take a liquid and put at a temperature and pressure where it would usually be a gas, but does not become a gas until disturbed. Supercooling is the equivalent for solids.
My question is, is it possible to both superheat and supercool a substance, meaning take a liquid substance, bring it to a temperature and pressure on its sublimation curve, yet maintain its liquid state.
If it is possible, what would happen if it were disturbed (like say by a loud blast of sound?)
 A: Yes. 
Here is a link to a phase diagram for water. It shows the temperatures and pressures where water is a solid, liquid, or gas at equilibrium. 
Ignore much of it. Look at the boundary between liquid and solid. If you are in the liquid region, and slowly reduce the temperature, you cross into the solid region. This usually means the water freezes. But it is possible to get supercooled liquid water. It freezes when disturbed. 
To be supercooled, the water has to be in the solid region. It cannot be superheated in this region. Water tends to freeze here, not evaporate. 
You could also change the pressure and can move across the boundary between liquid and vapor. Now it might be liquid that is superheated. It cannot be supercooled here. 
Right on the solid/liquid boundary, water can be either or or a mixture of both at equilibrium. If it is liquid, it isn't supercooled. It is just at freezing. 
The situation is similar for boiling water at the liquid/vapor boundary. 
You can start in the liquid region, and move to the solid/vapor boundary. If done carefully, this could be liquid water that would change to either a vapor, or a solid, or a mixture of both on being disturbed. 
