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In this paper we characterize topologically the empty interior subsets of a compact surface S which can be ω-limit sets of recurrent orbits (but not of nonrecurrent ones) of continuous flows on S. This culminates the classification of ω-limit sets for surface flows initiated in [Jiménez & Soler, 2001; Soler, 2003; Jiménez & Soler, 2004a, 2004b]. We also show that this type of ω-limit sets can always be realized (up to topological equivalence) by smooth flows but cannot be realized by analytic flows.