varnishlog, one of the tools provided with varnish cache, uses VSL Query Expressions (https://www.varnish-cache.org/docs/trunk/reference/vsl-query.html) to provide some powerful insights into the requests and responses.
Here is a how you can use varnishlog to show all client requests that are ending up with a 404 response.
sudo varnishlog -g request -i ReqURL -q "BerespStatus != 200"
Technically, this particular query shows all client requests with a response other than 200.
Breaking down the commands
-g request : shows all entries related to the request
-i ReqURL : forces varnishlog to only display the Requesting URL
-q “BerespStatus != 200” : query filter to only match non 200 responses. Note that the query has to be enclosed in “”.