If you can get the demolition started w/o doing much damage, if the flooring is installed on some sort of thin underlayment, you can rip it out along with the linoleum. You don't want to use any power equipment on the flooring because you can make the glue airborne. This is when the asbestos problem becomes a PROBLEM!
I cut the linoleum in a number of places until I found a 'seam' where 2 pieces of 1/4" plywood came together. I cut the flooring with a utility knife along that seam and them ripped up the plywood using a variety of tools. From there, I was able to rip out the rest of the floor. Of course, you can bend and fold the plywood like crazy to get it out the door. HOWEVER, if you're gonna put it out front for the trash guys to take away, most municipalities require that the 'sheets' be 4' x 4' or less. Just bend the sheets in half and cut the linoleum where you have a bend. Bend and break and cut as much as you need so all the pieces are 4' x 4', or less. Remember, DON'T use any power tools because of the potential asbestos hazard in the glue. I would advise that you wear a dust mask.
Another way to get the job started is to cut out a small area of linoleum with a utility knife, then, chisel out a hole in the underlayment. (This is assuming you HAVE an underlayment, AND, it's 1/4" (maybe 3/8") in thickness), and 'pry up' the underlayment to get you started. I've use an ice chopper to do the prying once I get it started with a claw on a hammer or a pry bar. YOu kinda just run the chopper under the underlayment and pry it up. It works great! You can use a large crowbar too to accomplish the same thing. Just cut the linoleum where seams from 2 separate sheets of underlayment meet.