There is already an active thread on the research Tu-144L and a simple search here or on google would provide info on it. I am not sure what other airframes are currently in a position to be made flyable again....others will know for sure.
hmmmm, the search function is not that reliable, but yesterday or the day before there was a lengthy thread on the "NASA supersonic test" flights. In any case, the plane is stored on the apron at Zhukovsly AFB on Ramenskoye Field, south of Moscow. I imagine it could be flown again without too much hassle.
I read that there were two or three Tu-144D's occasionally flying research missions in the early 90's, of which one was later converted to the Tu-144LL (flying laboratory). All remaining frames are presently stored near Moscow as Afay1 stated.