My Hero Academia might not be the saddest shonen out there, but there are characters with seriously miserable backstories. Many of these sad tales come from the villains, who found their way to villainy because of the way their lives went wrong. In other cases, it's the heroes who suffered.
While most of these examples have been covered in the anime, there are a few backstories that are manga-only, and even one from the third movie My Hero Academia: World Heroes' Mission. These include the origin stories of Dabi, Lady Nagant, and Rody Soul. If you're not caught up and want to avoid spoilers, go ahead and skip those entries.