Any suggestions on how to contain or kill off (they are taking over everywhere) blackberry vines?

How large an area are you talking about? You may be able to keep them in check by mowing frequently. We have wild blackberries growing at the back of our property, and they do spread slowly, but I haven't found them to be invasive.

They are actually shoots coming from vines in my neighbor's yard. They have tried to kill them off too but without success. Probably a 12'x6' area. I'll try the mowing suggestion. Thanks.