For a visitor the Marais is great, around the 4th. Take a look at rue Rambuteau. Very close to a lot of attractions - Notre Dame, the Pompidou, the river, a decent walk to the Musee du Louvre, a lot of people living in apartments so there are a lot of shops for daily needs.

For a Parisian, well, probably somewhere like the 19th or 12th (most of the people that I know who live in Paris are around these areas). 16th (Passy) if you are rich...but it's not a very exciting area unless you might enjoy the delights of the Bois de Boulogne.

Of the areas you mentioned earlier, Montmartre would be more typical of the two. But of course, everybody will have a different opinion. I much prefer the Right Bank (north of the Seine), but a lot of people will only stay on the Left Bank (south). And wherever you stay you're a quick metro ride to anywhere in Paris.