Stairs with more than four risers require a handrail. There are a lot of requirements that the handrail must meet to be considered safe and meet code requirements. Some general requirements (without getting into the exceptions to the rules) include:
1. There has to be a handrail on at least one side of the stairs.
2. The handrail must be between 34-38 inches from the stair tread nosing (leading edge) vertically to the railing.
3. The handrail must be "continuous" over the total run of the stairs (from landing to landing)
4. The ends of the handrail must return to the wall or terminate at newel posts.
5. There must be a minimum of 1-1/2 inches between the railing and the wall.
6. Most round handrails should be between 1-1/4 inches minimum width and 2 inches maximum width (Must be graspable). (There are lots of exceptions to this rule that allow for many other types of handrails.) For more information about handrails consult with your local building department.
Perhaps the most frequent defect found concerning handrails is that they do not return to the wall (with a close second being loose and/or completely missing handrails). When the end does not return to the wall a person's hands and/or belongings can "catch" on the ends of the handrail leading to falls and/or personal injury. I have known of persons trying to go down stairs in an emergency to dislocate their wrist when it caught in the opening. The left picture below is of particularly concern because: 1, it is brand new and someone should have known better, and 2, it is in a Condo stairwell, where the public is affected. The picture on the right is a handrail that is properly returned to the wall.
This next picture shows how a "difficult" set of stairs has been given a continuous handrail making many bends.
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WA State, Home Inspector Advisory Licensing Board