If there is grass on top of a block, it sometimes causes the block to become indestructible if you follow these steps:

1. Place grass on a block or find a block with grass on it.
2. (You must be in Survival mode) Start to break the block under the grass, but quickly move the cursor up into the grass's collision box.
3. It may take a few tries, but eventually you will get what is in the attached video.