For bicycles in California, this is the law: http://leginfo.legislature.ca.gov/faces/codes_displaySection....

    CODE TEXT
    VEHICLE CODE - VEH
    DIVISION 11. RULES OF THE ROAD [21000 - 23336]  ( Division 11 enacted by Stats. 1959, Ch. 3. )  
    CHAPTER 1. Obedience to and Effect of Traffic Laws [21000 - 21296]  ( Chapter 1 enacted by Stats. 
    1959, Ch. 3. )  

    ARTICLE 4. Operation of Bicycles [21200 - 21213]  ( Article 4 added by Stats. 1963, Ch. 479. )
  
    21202.  
    (a) Any person operating a bicycle upon a roadway at a speed less than the normal speed of 
    traffic moving in the same direction at that time shall ride as close as practicable to the 
    right-hand curb or edge of the roadway except under any of the following situations:

    (1) When overtaking and passing another bicycle or vehicle proceeding in the same direction.

    (2) When preparing for a left turn at an intersection or into a private road or driveway.

    (3) When reasonably necessary to avoid conditions (including, but not limited to, fixed or moving 
    objects, vehicles, bicycles, pedestrians, animals, surface hazards, or substandard width lanes) 
    that make it unsafe to continue along the right-hand curb or edge, subject to the provisions of 
    Section 21656. For purposes of this section, a “substandard width lane” is a lane that is too 
    narrow for a bicycle and a vehicle to travel safely side by side within the lane.

    (4) When approaching a place where a right turn is authorized.

    (b) Any person operating a bicycle upon a roadway of a highway, which highway carries traffic in 
    one direction only and has two or more marked traffic lanes, may ride as near the left-hand curb 
    or edge of that roadway as practicable.

    (Amended by Stats. 1996, Ch. 674, Sec. 4. Effective January 1, 1997.)

For motorized scooters, the only thing I could find is in the vehicle code: https://www.dmv.ca.gov/portal/dmv/detail/vr/scooters

    A motorized scooter is a two-wheeled device that has handlebars, a floorboard designed to be 
    stood upon when riding, and is powered by a motor.

    The Vehicle Code (VC) does not require registration, license plates to be displayed or the 
    scooter to be insured. Local authorities can regulate the registration for these scooters 
    pursuant to VC §21225.

    Even though insurance is not required, owners of these scooters should contact their insurance 
    company to determine if coverage is available.

    An operator of a motorized scooter must be at least 16 years old, possess a valid drivers 
    license or instruction permit, and wear a helmet.

    A motorized scooter may be operated on a bicycle path, trail or bikeway, but not on a sidewalk. 
    On the roadway, it must be operated in the bicycle lane, if there is one. On roads without 
    bicycle lanes, motorized scooters may operate where the speed limit is 25 mph or less, and shall 
    be ridden as close to the right hand curb as possible, except to pass or turn left. VC § 21226 
    (D) prohibits alteration of motorized scooters.