It is well known that the quality of stacking results (e.g., noise reduction, event enhancement, and continuity) can be greatly influenced not only by the traveltime operator chosen but also by the apertures used. We have considered two so-called diffraction-stack traveltimes, together with the corresponding apertures, designed to enhance reflections and diffractions, respectively. The first one is the common-reflection-surface (CRS) diffraction traveltime that is obtained from the general CRS traveltime upon the condition that the target reflector reduced to a point, which we refer to as the diffraction CRS (DCRS) traveltime. The second one is the double-square-root (DSR) traveltime, well established in time migration. We have observed that the DCRS and DSR traveltimes depend on fewer parameters (two in 2D and five in 3D) than the full CRS traveltime (three in 2D and eight in 3D). For the DCRS and DSR traveltimes, we have proposed specific apertures based on the projected Fresnel zone, which are able to produce high-quality stacked sections using less parameters to be estimated. The key factor in that approach lies in the choice of traveltime operators together with careful selection of stacking apertures. In particular, suitable choices of operators and apertures lead to stacking volumes in which reflections are enhanced (and the diffractions are attenuated) or the corresponding ones in which diffractions are enhanced (and reflections are attenuated). Synthetic and field data confirm the proposed approach has good potential for image-quality improvement.