Let and be closed, connected, and orientable surfaces, and let be a branched cover. For each branching point the set of local degrees of at is a partition of the total degree . The total length of the various partitions is determined by , , and the number of branching points via the Riemann-Hurwitz formula. A very old problem asks whether a collection of partitions of having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever is not the -sphere , while for exceptions do occur. A long-standing conjecture however asserts that when the degree is a prime number a candidate cover is always realizable. In this paper we analyze the question from the point of view of the geometry of 2-orbifolds, and we provide strong supporting evidence for the conjecture. In particular, we exhibit three different sequences of candidate covers, indexed by their degree, such that for each sequence:

• The degrees giving realizable covers have asymptotically zero density in the naturals.
• Each prime degree gives a realizable cover.