Answers

I would wait until the first frost, that very next day spray or paint leaves with Crossbow (if spraying isn't an option, as in beds with other plants or single shoots).  The sap in the stems will draw the herbicide down to the roots as the blackberry plant goes dormant in winter and kill it at the roots.  May have to repeat again the next year, but it has been very effective for us.    

Eat Them!!  

They have places that you can rent goats from.  They are prolific eaters. 

Dig it up and send it to me. I love blackberries.

I'm with you guys on the eating part! We had a bupper crop this year and the freezer is full. For those areas where we don't want them, there is product we found at Southern States that works. I bought it for our poison ivy but it ended up working on other things. The name escapes me, but it come in larger bulk containers and you mix w/ water.