The ISP is ZEN, and the modem connects to the router via PPPOE.

That much works.

I can ping IPs via both ethernet and wifi connections OK, but DNS doesn't work. Cannot ping any URLs - even ones definitely known to not be blocking pings.

It doesn't even work from the router itself, because it cannot access the Sonicwall registration site, saying that there is no DNS.

I found this
http://help.expedient.com/broadband/sonic_wall.shtml

which may be relevant, because (like that article) I am connecting to ZEN via the Draytek Vigor 120 whose ethernet interface is PPPOE.

Can anyone suggest a solution?

I've spent a few hours on it now. I got into the Sonicwall support site eventually, downloaded the latest (non beta) firmware (which was 2011 whereas the existing firmware was 2009; curious for a brand new router from Insight UK), installed that (which required a key to be obtained from Sonicwall... why do they secure their products like this???), reconnected to ZEN, but still cannot get any DNS.

I cannot see any appnotes on the web for the Vigor 120 and Sonicwall routers, either.

Previously, a Draytek 2900Gi worked fine with the 120, and works fine if I reconnect it in place of the TZ100. Looking at the config of the 2900 I cannot see anything which would be relevant, but the the TZ100 has 10x more config...

Update:

I got it finally working but there appear MAJOR issues with DNS over PPPOE, which the Draytek 2900 router doesn't suffer from.

I will send the Sonicwall TZ100 back - spent nearly 2 days on it. It's a piece of junk and the company's extreme policies on support and even registering the actual router on their system before "full functionality" is enabled is ridiculous.