I will say, screen printing on a striped or patterned shirt is really not that common. BUT, if you do decide to go this route, make sure you think about how the lines of your imprint will look against the lines of the pattern.

Stripes aren't always perfectly level on a T-shirt, and an imprint with a straight edge will only highlight those inconsistencies. In the example above, the blue rectangle of the Old Bay logo is perfectly level, but it looks crooked because of the surrounding stripes. The easiest solution, of course, would just be to print on a solid color shirt. If the stripes have to stay, the next best option is to back the logo with a circular shape. This way, the misaligned stripes are less noticeable.