Homes between 500 SF and 2500 SF are designed to a large degree around the stairs. The reason for this is that the stairs tend to dictate much of the circulation pattern in the floor plan, on both floors. If I place the stairs in the wrong place, it seems impossible to find a room plan that works well. A lot of space can be wasted if it is placed poorly.
A linear stair case on a 9 foot climb extends about 11 feet, but it requires a landing which extends that to 14 feet. Then it requires a 14 foot hallway to walk from the top of one flight to the bottom of the next. This hallway is the wasted space. So this 8ft x 14ft space uses 112 square feet.
The switchback stair case, or the switchback with a winder, will only require 84 square feet, saving 28 SF over the Linear stair. You still need the landings, but you don’t need the hallway. There are three walls to the switchback stair that can all be in rooms being useful. If you add the winder at the turn, the switchback will only require 76 SF. These are best used in very small homes when every inch counts.
A “spiral” staircase is usually a grand and decorative object built on the site with extra craftsmanship. Most people use the word “spiral” to refer to a “circular” staircase, and could also be called a helix, which is usually prefabricated in a metal shop and then shipped and installed on the site. The national building codes do not permit a metal “circular” staircase to be a primary staircase, unless it is 36 inches wide and the railing ballasters are no greater than 4 inches apart. This is large and rare.
In larger homes, when space is not an issue, any staircase style can usually be designed to fit well.