The resonant vibrations of small (microliter) sessile water droplets supported on solid substrates were monitored using a simple optical detection technique. A small puff of air was used to apply an impulse to the droplets and their time dependent oscillations were monitored bypassing a laser beam through the droplet and measuring the variations of the intensity of the scattered light using a simple photodiode arrangement. The resulting time dependent intensity changes were then Fourier transformed to obtain information about the vibrational frequencies of the droplets. The resonant frequencies of droplets with masses in the range 0.005-0.03 g were obtained on surfaces with water contact angles ranging from 12 +/- 4 degrees to 160 +/- 5 degrees. The contact angle dependence of the resonant frequency of the droplets was found to be in good agreement with a simple theory which considers standing wave states along the meridian profile length of the droplets.