Let w(z) be a meromorphic algebroidal function for |z|<+∞. If limr →∞lg T(r, w)/lg r =0, limr →∞lg T(r, w)/lg lg r =+∞, then there exists a direction arg z=θ0 such that, for any N>0, ∑i1/(lg|zi(a; ?(θ0, δ))|)N=+∞, with 2 v possible exceptions for a. If limr →∞T(r, w)/lgKr =+∞, limr →∞lg T(r, w)/lg lg r =M, then there exists a direction arg z=θ0 such that ∑i 1/lg|zi(a; ?(θ0, δ))|)σ=∞ (σ=M-2 or σ=M-2-ε), with 2 v possible exceptions for a. For meromorphic functions, these singular directions have not been studied yet.