Destructive 3 year old or merely curious how things work?
My son is very destructive. I work nights so I sleep on and off through out the day. His destructiveness isnt often but he's managed to break his bed frame 3 times, he pulls the fluff out of pillows and tears pages out of books. I put him in time outs and explain why what he did was wrong but it doesnt always work. He tends to do these things when he wakes up and he's alone in his room. Are my hours causing his behavior? Is it a need for attention? Is he just curious and it goes wrong?
First, please note that a child of three years of age should not be left unattended while he is awake. Children of his age will be curious which may lead to an unsafe situation.
Consider child care/ preschool, a babysitter (even a teen who has taken a babysitting course), a parent (swap child care perhaps) or a relative? This way you can get the rest you need and your child can be cared for and given the kind of developmental stimulation he is seeking. Be sure that the person watching your child will provide him with activities and not just place him in front of a TV or computer screen as this is not ample enough activity for a child of three. (In my humble opinion)
I have a friend who works night shifts, he has a young son also and does the same as you sleeps on and off through the day, his child is the same. What I suggest is, your son is looking for a little attention, personally I think young children need stability and discipline. Working the hours you do would definitely have an impact on your child's needs. I recommend sourcing a babysitter. Children who get lack of attention seem to become destructive as I've experienced first hand. I hope you find a solution soon.