Which cooking oils can be used for frying to produce the crispiest and least greasy foods. Do different foods require different oils? If so, what is best for battered foods, and which is best for non battered foods *potatoes, chicken wings, etc.)
The desired texture of 'crispness' is the result of many factors, and oil choice is generally not the biggest factor. Consider a french fry. The generally accepted best practice for fries is a blanch and finish method. This allows the fry to be cooked throughout and have a crisp surface, without being overdone and leathery. To accomplish this, the finish frying is generally done at a higher temperature than the initial blanching.
This blanching step can be done, and is easier to do at home, in a different liquid. I often simmer chicken wings in a flavorful stock, with some aromatics. You then want to cool, and especially if cooking in water, dry the food. This allows the par cooking to finish. Now, when finished from this cooled state, the outside can crisp without overcooking the inside.
I use this practice when preparing food at festivals, and it is the common practice in most restaurants to prepare crisply fried food.
The final crisping of potatoes and the skin of chicken comes from driving the water out. (This is what produces the bubbles) Here temperature is key, as the temperature must be high enough to turn the internal water into steam, but not too high to toughen and overcook.
All that being said, different cultures use a variety of oils for essentially the same effects. Crispness can be accomplish using any number of oils. The practicality is about cost, smoke point, adding unwanted flavors, and dietary concerns. There is a nice chart of smoke points over here. Note ghee (clarified butter) has a high smoke point and would impart amazing flavor (assuming you like butter) but would be expensive to fry in.
The crispness then is generated by relatively high heat, so par cook chicken wings and potatoes, etc. Cut battered veggies thin, so they can cook quickly. Choose a fat that you like and can afford, and use technique to get the best crispness.
As for greasiness, this often goes hand in hand. Greasiness, or amount of grease left on the food after cooking is temperature dependent and depends on crowding and draining after cooking.
If food is cooked at a lower temperature, then expelling of water is less vigorous. As you remove the food from the fat, the oil has a better chance of clinging if the steam moving out was less volume.
Crowding is important here. If there is too much food in the frying vessel, then the space between pieces of food turns into a steam area, instead of hot oil area. Now food is cooking in a moist ~212 F environment instead of the oil temperature. Those areas will not be hot enough to get crisp or expel extra oil and lose the greasiness. Good rule of thumb is you should be able to easily stir and move the food around while frying.
Finally, draining. Have a rack or platter with absorbent paper. Remove small amounts quickly. If using a 'spider' you can use an up and down motion to shake a little excess fat off. Now place in a single layer to avoid steaming. These final steps help, but if the food comes out flabby and soaked in grease already, they won't fix anything.
Seeing that crispness has been addressed in another answer, I would like to add some points on greasiness. The best solution is to roast the vegetables without oil (in contrast to grilling or stir frying) and optionally add oil after the process. This is typically how Mediterranean veggies are prepared. This can be achieved using an oven rack. Alternatively, a cast iron frying pan can be used. Non-stick and steel pans are not suitable.
To reduce the oiliness of deep fried food, the temperature has to be within a certain range, neither too hot nor too cold. Once fried, the food can be placed on a plate with absorbent kitchen tissues. This will soak up some of the excess oil.