I admit that this is probably the consequence of living in a less populated state, but when they ask the nature of your emergency here (NM), the dispatcher typically makes a decision of what response is needed.
All three are sent for most vehicular accidents for traffic control (state police, county sheriff, or city), fire suppression (volunteer FD, usually), and medical services (sometimes volunteer, especially in the county). Judging by the comments below, I suspect there's an advantage to living in an area that doesn't have tax money to blow. ;)