The last time I did a job like this I was dealing with 15"-20" stumps cut at grade. I used my 580 Case and determined on which side of the stump to dig. First, I went to the opposite side and cut a ditch 2'-3' deep to remove any surface roots. Then I repositioned my machine and dug on both sides of the stump, forming a V. At 5'-6' I used the hoe to "rock" the stump and continued digging as necessary. On some of the 20" stumps I wound up going to 8' before I could pull them out. Even with backfilling and grading between each stump it was a slow and frustrating job that could have been avoided if the property owner had left 6' above grade so I could have used a heavy dozer.
As for blasting, regulations vary. Check to see what your state and local laws require, and if you need a license to purchase and use. Of course you need to place the charge correctly which requires boring a hole. When done correctly, the sound will be minimal and blast will only loosen the stump making machine removal far easier. Unlike using a machine, if blasting is done incorrectly you may not have a second chance.