One thing to note about DNS servers and providers is that they aren't always trustworthy, not even if they're Google or your favorite ISP. Any DNS server can compromise your privacy, and they are likely tracking your browsing habits and keeping logs of it. Sometimes it's for Marketing/Research purposes such as Google's GMail service which they admit is scraped/datamined. I would expect nothing less from their DNS service.
The other danger with such widely used and publicly advertised DNS services in general is that if they are ever attacked and poisoned it could result in ID theft for many people. Eg. if Google's DNS cache is poisoned or servers hacked, and someone sets paypal.com to resolve to a hacked server, those people will unwittingly give their details away to the phishing site, or worse yet, if they login to their bank account and the cache is poisoned to a hacked server for a major bank.
As for ISPs, I would not trust them more either, of course your own ISP probably does the same but do keep this in mind that many DNS providers don't have good intentions. At the very least, remember that your browsing habits and websites and visited are can always be tracked and monitored by whoever your ISP is and whoever your DNS server operator is.
The one benefit of using these servers is that sometimes they can help you can.
It can make tracking and logging by your own ISP more difficult, they at least can't do it at the DNS level of their own servers unless they are actively logging all traffic outgoing to udp port 53 (DNS).
In countries where content is banned, sometimes you can get around filtering and blocks by simply using a foreign DNS server (assuming those servers aren't blocked and assuming the block is at the DNS level only and actively filtered by port filtering and content inspection, or based on IP etc..).
With the warning/disclaimer above here are some free DNS server providers.
Verizon Free DNS Server (used by many VPS server providers)
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