A CRJ flight crew suffered an NMAC with a private high winged aircraft while in VMC. TCAS TA and RA advisories contributed to a safe outcome.

Narrative

I was on initial climb out and the First Officer was the flying pilot. We received a turn to the north over the ocean at which time we were assigned 11;000 MSL. ATC notified us of traffic about our 11 to 12 o'clock position. I believe it was stated that 'altitude was unverified' by the Controller. During the outside scan I noticed a small high wing single engine aircraft and noted him to be level at about 10;500 MSL. The aircraft appeared very small and a reasonable distance away. I returned my scan back inside the aircraft to check the TCAS system at which time a Traffic Advisory sounded on the TCAS. Our aircraft was approaching 10;000 MSL at that time. Very soon after the TA sounded the pilot flying disengaged the autopilot and responded to a Resolution Advisory. The audible 'Monitor Vertical Speed' put us into an approximate 3;000-3;500 FPM climb until reaching an altitude of 12;000 MSL where we received the 'Clear of Conflict' advisory. I was able to look out the left side window in time to see the aircraft fly under the nose of our aircraft. I estimate we missed each other by 300-500 FT. During the RA I announced several times without acknowledgment from ATC; that we were responding to an RA. There was heavy radio traffic and I may have been stepped on and unheard. Once at 12;000 MSL ATC asked us to 'confirm level at 11;000'. I responded once again 'No! We are responding to an RA'. He then requested that we let him know when we were complete with our actions. I responded with 'we're done and are at 11;600 descending back to 11;000'. The flight continued without further interruptions. There is some confusion as to whether or not the VFR traffic was talking to any ATC facility. If the pilot was not speaking to a controller then surely that is a mitigating factor.