Believe I'd notice in the data graphs if my anemometer showed no wind and no change of wind direction for a period of time; for sure for a month. 
Well, as it turns out, it wasn't THAT bad. Not sure how I was looking at wunderground, but looking at my raw data, it only showed about 16 hours of no wind. So some spider had a fun night binding up my anemometer.
The problem is, I don't pay THAT close attention to my weather station. I really only noticed because I was in my yard watering plants and noticed the cups not spinning. =/
As to how to implement, I would assume you would have two things you could set in preferences. Value to check against, and Duration of not changing. So, in the case of wind speed, the value would be 0 (units), and after X hours/X days, if it doesn't change (or rather, if the max wind is 0.0 for several days), then there's probably a problem. With temp, humidity, etc, you'd actually have to have it check multiple days and see if the value never changes, rather than have it be enterable (I doubt most stations fail in such a way as to reliably say "0 degrees" or "100 degrees", I'm sure they stick on some undetermined value.