Recently I've started studying the theory of singular values for entire functions so I'm far from being a specialist in this field. In the literature I came across the following results:
In  Gross constructed an entire function for which every point in the complex plane is an asymptotic value.
In  Heins proved that every Suslin analytic set is an asymptotic set of some entire function.
In  Heins proved that for every countable set A in the complex plane there exists an entire function whose set of locally omitted values coincides with A.
(Q1) My question is, are any of these examples locally univalent functions?
Unfortunately I was not able to varify these things by myself since I don't read German  and since I was not able to find an article  in our library basis or online.
Regarding the third result I think that the answer is positive, since the function is obtained as an associated mapping function of a parabolic surface spread over the sphere with only logarithmic ramification points. Is that correct?
(Q2) Do you know any (other) references with a construction of an entire function having a prescibed singular set - closed subset of a complex plane (I've seen the paper of Bishop, Constructing entire functions by QC-folding, where this is done)
 W. Gross: Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzwert ist (German). Math. Ann. 79 (1–2), 201–208 (1918)
 M. Heins, The set of asymptotic values of an entire function, Tolfte Skandinaviska Matematikerkongressen (Lund, Sweden, 1953), Proceedings of the Scandinavian Math. Congress,Lund, 1954, pp. 56–60
 M. Heins: Asymptotic spots of entire and meromorphic functions. Ann. Math. (2) 66, 430–439 (1957)