I think a good agent will search a variety of ways and much of that depends on the buyer and their priorities. Often times, it takes some dialog with the buyer, or even showing them some homes to see what they love or not in order to find out what is important to them. Sometimes the things they say are important fall further down the list when other things move their way UP the list. So the search parameters depend on the needs of the client.
But as far as the actual search, I start with a search of an MLS area by number but then also will search by school district, or lot size or age of home. I usually use a map search (the rubber band search) when a client needs to be within a certain distance of something specific like a relative or perhaps their workplace.
Spelling out the city may leave some homes out of your search because there may be variations in how the name is entered. For example: St. Charles, Saint Charles, St Charles, StCharles...you get the point? But if I enter 60174 in the zip code field (not the area) I will get only homes in the "incorporated" area of St. Charles. If I enter 60175 in the zip code I will get unincorporated St. Charles. These are nuances of the MLS that most good agents are familiar with.
Another important note, be sure your listing agent is accurate in entering all the information in the MLS in order for your property to be returned in the search results. Even in the remarks section be sure your agent includes key features (or key words) of the property because many agents will search the remarks for key words like; "screened porch" or "perennial" garden. Accurate spelling is important too. If your agent misspells perennial (and many do!) it won't come up.