Here`s a website with some info & history of Army shoulder patches, & distictive unit insignia:
American Military Patches, Other Insignia and Decorations
Basicly, shoulder patces were authorized for only Division level & above & only 1 could be worn on the sleeve. Therefore a unit would wear the patch of the command unit they were assigned to. For example, all componet units of the 3rd Infantry Division would wear the Blue & White Stripe division patch. As Marion mentioned, the "bastard" units such as many engineer units even tho they were under Corps Command but usualy operating in divison areas would just wear their regimental or battalion DUI. Corps and Army patches were mosty worn by HQ staff officers & other personel ie: Intel, Signals, MP etc working only at the Corps or Army HQs and command posts.