Survival rates - mortality rates in biology, and breakdown/failures in machines
Reliability analysis - how long can a machine be expected to perform for a particular level of performance
Wind speed - used by wind power industry

Hydrology - extreme events like big spikes in daily rainfall/discharges from a natural water source

Insurance - Size of Reinsurance claims

Technological change - Sharif - Islam model

What exactly are you studying, and what is the data that makes you feel that Weibull is a better distribution to model, then others?