[Part I is reviewed above (see Zbl 0631.03031).] An r.e. \(tt\)-degree is called singular if it contains exactly one r.e. m-degree, and a \(T\)-degree is called singular if it contains a singular r.e. tt-degree. Singular degrees were first constructed by \textit{A. N. Degtev} [Algebra Logika 12, 143-161 (1973; Zbl 0338.02023)]. We show \(\underset\sim 0'\) is singular, singular \(T\)-degrees are dense in the r.e. degrees, but there are nonsingular r.e. \(T\)-degrees.