Can you synthetic oil in a Honda shadow 750?

Yes, unless it contains a friction modifier which would be indicated by the circle on the container. It should say something like " Energy Conserving". If it does, stay clear of that particular oil as friction modifers can cause your wet clutch to slip. One of the most popular synthetic oils used in the Shadow line is 5/w40 in the Rotella line of diesel oil. It comes in the blue gallon jug and is readily found at Wal Mart and other discount stores.