Safest options...staying in the suburbs like Brentwood/Franklin, Hendersonville, Mt. Juliet, or Bellevue would be the safest options. However, mlp206 is right, East Nashville/Gulch/Vanderbilt/12South are all popular for singles. Of course, you often pay for the popular places. East Nashville and 12 South are transition areas with some definite crime issues although personally I really like both areas and would consider moving into them...especially East Nashville.
Honestly, I would give Bellevue a close look. It is a good area, safe, affordable, and there seem to be a large number of 20-30 somethings in that area.