While I like @Jigar Joshi's answer, I'll provide some American Connotation, if I can. (Caveat, not a professional culturist)

Shop

I would say that a shop is somewhere you go to have a service preformed, or only provides a single or narrowly focused good. Usually contains a workshop of some kind where the goods are made on-site for the customer.

• Body Shop refers to an auto repair facility (autobody shop)
• Wood Shop refers to a place where you can go to get custom woodworking done
• Coffee Shop a place you can go to get hand-made drinks
• Butcher's Shop a place to get custom cuts of meat

Store

I would say that a store is the more common American term for a place you go to buy something. They usually have many different sections and offer a wider variety of goods than a shop. The goods you purchase are usually made off-site and only stored on location.

• Department Store buy furnishings and clothes for each room of the house
• Grocery Store buy food and many other consumables
• Online Store buy anything! :D

Shoppe

Generally a fancier term, as @Jigar Joshi mentioned, to give an air of authenticity and aristocracy.

• Coffee Shoppe here you can buy even more expensive fancy coffee.

But where ever you go, you'll end up shopping regardless of the store.

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.

In England and Australia a shop is what North Americans refer to as a store. A shop is a place for conducting retail business.

As pointed out above, some retailers may brand their shop as a Megastore because the term store has become trendy in advertising but it is nevertheless still a shop.

Store only applies to a 'department store' (a shop with several departments) or a 'general store' in a village or small community which sells a variety of essentials.

Another international curiosity: I notice in the spread of English in German websites, they have online shops rather than stores.

In American English:

Store - A place to purchase things.

Shop - I think primarily is used as a short form of 'workshop'. I don't hear it being used very often as a synonym of 'store', although it is a valid usage.

Shoppe - Used on stores to make them sound medieval.