Why is she so often pictured with a shovel? I assume it's to bury the people she maniacally kills when their socks don't match, but I don't think it was shown or alluded to in the series at all. Is it from the manga?

A small explanation about that, taken directly from wikipedia [lol]

"her favorite weapon of choice is a spade and the kanji of her first name can be joined together to form the first character of "to bury" (埋める, umeru?)"