It really depends on who is fixing the veggies. If its me, the stumps (love that too ) will get chopped up into whatever we're having, soup, salad, etc. If its DH, I grab them from the counter before he can toss them on the compost & chop them all up into my dish
I use mine in my homemade dog food. The recipe is 1/3 meat, 1/3 starch and 1/3 veg. I throw whatever broccoli stems I have in the fridge into the mix. My dog seems happy with it!
Sure, they're great in stirfries (and those who have eaten inexpensive Chinese restaurant food have undoubtedly eaten them already in various dishes where the veggies are diced or in those little sort of square-triangle shapes, where they are quite commonly used). Somewhere between being liek a water chestnut and a bamboo shoot in texture