What your child is doing, this stimming, is also called Perseveration. It's actually very beneficial to your child. I know that it can get annoying, especially the verbal stimming, ESPECIALLY when it's echolalia. It can take the form of physical stimming, such as spinning, tapping, rocking, flapping, hopping, jumping, repeating words or phrases, various repetitive behaviors, etc.
Stimming, ie Perseveration can:
Help people with ASD focus on a task.
Help them self-soothe.
Self-stimulate... sometimes beneficially, sometimes not.
Help organize thoughts.
Help as a coping strategy.
Help release nervous energy.
Some people believe that stimming isn't beneficial because it can over-excite a child or inhibit them. It's also sometimes believed that stimming reinforces only these behaviors and keeps a child from concentrating and learning other techniques that are more beneficial forms of self-soothing.
The biggest problem from stimming seems to really be the concern that it's stigmatizing socially. The fact is that EVERYONE stims. Do you ever tap your pencil or shake your leg nervously? Do you chew on pen caps? Twirl your hair around your finger? Hum a favorite tune when writing a letter or blog? Those are all stimming.
But we want our kids to fit in, so most often parents try to "stop stimming." The problem is you can't stop one behavior without replacing it with another. If you want him to stop the humming and singing, because it's disruptive to other students, try to limit it to safe zones in the home. Give him permission to do it in his bedroom and the bathroom. But try to help him find something he can replace the humming and singing with when he feels the need to stim at school and in public, and practice it with him in the common areas at home. Maybe put a special bracelet on his wrist that he can touch occasionally or something else discreet. His therapists at school should be able to help find something and incorporate it into his IEP.