A variety is called tabular (T) whenever it is generated by one finite algebra. A pretabular (PT) variety is a non tabular variety whose proper subvarieties are all tabular. Those notions were firstly studied for Heyting algebras, in the sixties and seventies. An important role of PT varieties is to check if the tabularity problem (i.e. deciding if a certain variety is tabular or not) is decidable. In a recent work we started studying T and PT varieties of MTL-algebras, finding some preliminary results. In this paper we take a step further, by studying the decidability issues for some properties for tabular varieties of MTL-algebras, like tabularity, consistent tabularity, non-boolean tabularity, single chain generation and amalgamation property. We study the decidability and the computational complexity of those problems.