We have made polarimetric observations of three Type Ia supernovae (SN Ia's) and two Type II supernovae (SN II's). No significant polarization was detected for any of the SN Ia's down to the level of 0.2%, while polarization of order 1.0% was detected for the two SN II's 1994Y and 1995H. A catalog of all the supernovae with polarization data is compiled that shows a distinct trend that all the five SN II's with sufficient polarimetric data show polarizations at about 1%, while none of the nine SN Ia's in the sample show intrinsic polarization. This systematic difference in polarization of supernovae, if confirmed, raises many interesting questions concerning the mechanisms leading to supernova explosions. Our observations enhance the use of SN Ia's as tools for determining the distance scale through various techniques but suggest that one must be very cautious in utilizing Type II for distance determinations. However, we caution that the link between the asphericity of a supernova and the measured "intrinsic" polarization is complicated by reflected light from the circumstellar material and the intervening interstellar material, the so-called light echo. This effect may contribute more Substantially to SN II's than to SN Ia's. The tight limits on polarization of SN Ia's may constrain, progenitor models with extensive scattering nebulae such as symbiotic stars and other systems of extensive mass loss.