You could make them into soup.
Fry some finely chopped onions in a tiny bit of olive oil until soft. Add in the asparagus trimmings. Cook for just 2 mins. Add chicken stock. Season. Blend. (I add cream at the end, but I'm addicted to cream)
There are signs up all round our area: 'Asparagus for sale', 'New spuds', 'Strawberries'. Little stalls everywhere, plus all the local farmers. Cromer crabs are on offer too! Gorgeous.