For a Z-graded Gorenstein ring R, we study the stable category of Z-graded maximal
Cohen-Macaulay R-modules, which is canonically triangle equivalent to the singularity
category of Buchweitz and Orlov. Its thick subcategory \underline{CM}^Z_0R is central in
representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split
triangles. In the case dim R = 1, we prove that \underline{CM}^Z_0R always admits a silting
object, and that it admits a tilting object if and only if either R is regular or the a-invariant of R
is non-negative. We also show that, if R is reduced and non-regular, then its a-invariant is
non-negative and the above tilting object gives a full strong exceptional collection.