The question is a little open, but I'm wondering how many TCAS resolutions (i.e. the serious type) occur every week or month over sizeable region like the United States or Europe.
I found a very interesting paper from 2007 that analyzed TCAS performance. They analyzed recorded data to come up with 1 RA per 116 flight hours.
Over this time period, monitoring took place for approximately 190 days, and roughly 200,000 Mode S flight hours were observed within the sensor’s 60 nmi coverage area. We observed a total of 1725 RA events, corresponding to an average of 9 RAs per day, or approximately one RA every 116 flight hours.
They explain that the data was taken in the Boston area, so the high amount of traffic contributes to the high number of RAs.
This thread on Airliners.net suggest differently, at least in the sense that for those pilots it is a matter of a few over years, depending on where they typically fly.