Proximity-based mobile social networking (PMSN) refers to the social interaction among physically proximate mobile users directly through the Bluetooth/WiFi interfaces on their smartphones or other mobile devices. It becomes increasingly popular due to the recently explosive growth of smartphone users. Profile matching means two users comparing their personal profiles and is often the first step towards effective PMSN. It, however, conflicts with users' growing privacy concerns about disclosing their personal profiles to complete strangers before deciding to interact with them. This paper tackles this open challenge by designing a suite of novel fine-grained private matching protocols. Our protocols enable two users to perform profile matching without disclosing any information about their profiles beyond the comparison result. In contrast to existing coarse-grained private matching schemes for PMSN, our protocols allow finer differentiation between PMSN users and can support a wide range of matching metrics at different privacy levels. The security and communication/computation overhead of our protocols are thoroughly analyzed and evaluated via detailed simulations.