Infants
Display a purchased infant's mobile as directed on the packaging. The headboard of the crib is a good spot to mount the bracket, as long as the baby is unable to stand and reach it. Crib side rails are another option. Mobiles that suspend from the ceiling are too far away for babies to see until their vision develops further. You want to be sure to well secure the mobile to either mounting surface.
Preschool Aged
Mount the mobile by attaching it to the ceiling using a cup hook. Once a baby can stand, walk and reach for things, a mobile that hangs near at hand can be dangerous. Even safely-constructed mobiles get broken when left within reach of a toddler. A mobile that doesn't hang very low and is lightweight is a reasonable option for a toddler's room.
School Aged
A mobile created to stand upright is ideal for this age group. Free-standing mobiles sit upon a base and pivot, requiring no hanging or mounting. The mobile still needs space to turn freely. The top of a bureau or dresser is ideal if the span of the mobile is less than double the breadth of the dresser top.
Teenaged
Use a molly-bolt, anchor bolt or toggle bolt to hang heavier mobiles -- ideal for this age group. An eye screw or cup hook is sufficient if the mobile is very light in weight. Read the directions that come with the mobile for instructions and suggestions for hanging. A drill or punch awl, as well as a screw driver, may be required for installation. Large, heavy mobiles are ideal for decorating the teenage child's room.