Earlier, we considered an initial-boundary value problem for a one-dimensional Boussinesq-type equation in a domain that is a trapezoid, in which the theorems on its unique weak solvability in Sobolev classes were established by the methods of the theory of monotone operators. In this article, we continue research in this direction and study the issues of correct formulation of the boundary value problem for a one-dimensional Boussinesq-type equation in a degenerate domain, which is a triangle. A scalar product is proposed with the help of which the monotonicity of the main operators is shown, and uniform a priori estimates are obtained. Further, using the methods of the theory of monotone operators and a priori estimates, theorems on its unique weak solvability in Sobolev classes are established. A theorem on increasing the smoothness of a weak solution is established. In proving the smoothness enhancement theorem, we use a generalization of the classical result on compactness in Banach spaces proved by Yu.I. Dubinsky ("Weak convergence in nonlinear elliptic and parabolic equations Sbornik: Mathematics, 67 (109): 4 (1965)) in the presence of a bounded set from a semi-normed space instead of a normed one. It is also shown that the solution may have a singularity at the point of degeneracy of the domain. The order of this feature is determined, and the corresponding theorem is proved.