I often use a plastic onion bag filled with seeds or buy the bell shaped mesh covered nyger seed to hang in my trees for the birds. Is this safe for them?

No, they are not safe for birds. On multiple occasions at Le Nichoir I have received birds entangled in these mesh bags. This past winter a Black-capped chickadee was brought to the Centre with its leg amputated from being stuck in a soft plastic mesh feeder. These feeders are dangerous and should not be used. Consider using solid plastic or metal small seed feeders instead. They not only are safer but are more durable and can be reused.