Weather Push Alerts
The weather push alerts on the smartphone (assuming you are referring to emergency alert system all smartphones come with, along with the Amber alert system) ONLY goes off if the threat is severe enough, such as Tornado Watch or Warning. However, my smartphone's emergency push alert never activates for severe thunderstorm warning or snowstorm warning.
But, WeatherAlertUSA seems to work very well. It was a little tricky figuring out how to set it up, unneccesarily complicated website, but I got it working. I am sure there are also weather apps that send push alerts to your phone as well, but I am not aware of any good ones. I also use the weather underground app, which seems to be even more accurate and up to date than NOAA, but still doesn't send me push alerts for some reason. I have to go into the app itself to view any warnings or alerts.