We say that a Cohen-Macaulay local ring has finite CM+-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite CM+-representation type from various points of view, relating it with several conjectures on finite/countable Cohen-Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM+-representation type are exactly the local hypersurfaces of countable CM-representation type, that is, the hypersurfaces of type (A∞) and (D∞). We also discuss the closedness and dimension of the singular locus of a Cohen-Macaulay local ring of finite CM+-representation type.2010 Mathematics Subject Classification. 13C60, 13H10, 16G60.This theorem may look technical, but it actually gives rise to a lot of restrictions which having finite CM + -representation type produces, and is used in the later sections. One concrete example where Theorem 5.5 applies is when I = (x) and (0 : x) = (x); see Corollary 5.9. Here we introduce one of the applications of the above theorem. Denote by CM(R) the category of maximal Cohen-Macaulay R-modules, and by D sg (R) the singularity category of R. Theorem 1.5 (Theorem 5.8). Let R be a Cohen-Macaulay local ring of dimension d > 0. Let I be an ideal of R with V(I) ⊆ V(0 : I) such that R/I is maximal Cohen-Macaulay over R.Suppose that R has finite CM + -representation type. Then one must have d = 1. If I n = 0 for some integer n > 0, then CM(R) has dimension at most n − 1 in the sense of [18]. If R is Gorenstein, then R is a hypersurface and D sg (R) has dimension at most n − 1 in the sense of [23].There are folklore conjectures that a Gorenstein local ring of countable CM-representation type is a hypersurface, and that, for a Cohen-Macaulay local ring R of countable CM-representation type, CM(R) has dimension at most one. The above theorem gives partial answers to the variants of these folklore conjectures for finite CM + -representation type.