Depending on your quilt design, determining the straight of grain before cutting your pieces can be important. In general, if you’re worried about edges or even whole blocks stretching on you, using the straight of grain can add stability as it is less stretchy than across the grain.
For borders, you can reduce the chance of getting a wavy edge to the quilt by cutting the border fabric on the straight of grain. Measure the total lengths needed, cut, then mark where the blocks fit and pin. This works for sashing too and is especially helpful when the edges of the blocks are on the bias.
Using a directional print or weave in the border may mean compromising stability for the effect you want to achieve. A work-around is to add a narrow inner straight of grain border in a solid or small-print fabric to stabilize both the main border and the centre of the quilt.
Sometimes you want the stretch you get from having as much of the piece cut on the bias as possible—think bias strips and how easily they curve into loops. Many appliqué pieces are also easier to nudge into place and adjust on the fly when their edges have the ‘give’ that being on the bias provides.