The integral equations which are characterized by singular integral operators with shift appear frequently in a large variety of applied problems (we refer to [KaSa01, KrLi94] for a general background on these operators and historical references). Thus, it is of fundamental importance to obtain descriptions of the invertibility characteristics of these operators. Although some invertibility criteria are presently known for several classes of singular integral operators with shift, the corresponding criteria still remain to be achieved for many others. In addition, among all the classes of singular integral operators with shifts, the ones with weighted shifts typically reveal extra difficulties.