Checking the IP address of the DNS resolver in use :
These servers will reflect the IP address of the host querying them.
dig whoami.akamai.net +short 18.104.22.168 dig whoami.ultradns.net +short 22.214.171.124
Checking if your own local DNS resolver is open :
This one is pretty much self-explanatory.
dig amiopen.openresolvers.org TXT +short "Your resolver at 126.96.36.199 is CLOSED"
Checking your resolver's source port behavior :
A test server allowing to check resolver's source port randomization, and grading them (either POOR, GOOD, or GREAT randomness). More information here.
dig porttest.dns-oarc.net TXT +short porttest.y.x.w.v.u.t.s.r.q.p.o.n.m.l.k.j.i.h.g.f.e.d.c.b.a.pt.dns-oarc.net. "188.8.131.52 is GREAT: 71 queries in 37.0 seconds from 71 ports with std dev 19761"
DNS Reply Size Test Server :
A test server allowing to identify resolvers that cannot receive large DNS replies. More information here.
dig rs.dns-oarc.net TXT +short ;; Truncated, retrying in TCP mode. rst.x4091.rs.dns-oarc.net. rst.x3837.x4091.rs.dns-oarc.net. rst.x3843.x3837.x4091.rs.dns-oarc.net. "184.108.40.206 DNS reply size limit is at least 4091" "Tested at 2013-05-16 18:00:19 UTC" "220.127.116.11 sent EDNS buffer size 4096"