With walnut shavings as with all juglans, I toss them. At the price
of walnut around here though, that usually isn't a problem.
If they are planer shavings, I save cherry, oak and maple for grilling
flavor or for a short barbecue. End pieces, small drops, etc. all go
into the BBQ pile as smoking wood.
It there is a lot of shavings, I use them for compost in my herb
garden, or with a bunch I try to get them on my tomatoes as compost.
If it is sawdust from the saw, I save as much as possible for
refinishing work. For the old fashioned method of using he sawdust as
a mild abrasive, cleaner, and liquid absorbent good sawdust from a saw
can't be beat.