I'm presuming you're referring to these words used as nouns to mean a mercantile business of some sort. Historically, stores were businesses which kept inventory to be sold and shops where businesses where items were manufactured or repaired. Nowadays, shop and store are generally synonymous, but each has slightly more typical uses depending on what is being sold.

In my part of the world (Canada) you would generally refer to most commercial places of business as a store (grocery, convenience, clothes, book, furniture) while shops are reserved for repair, flowers, or gifts (gift shop is quite idiomatic.) Thus, you would typically hear bicycle shop, rather than store, because repairs are done there. It would not sound that strange to hear shop when referring to clothes, book or furniture stores, however.

Shoppe is an archaic spelling of shop and is used only in proper names of places wanting to sound quaint and old-fashioned. The Pop Shoppe and The Medicine Shoppe are a couple canadian examples.