In order to mitigate the "Poodle" vulnerability, I'd like to disable SSLv3 support in my (in this case, TLS, rather than HTTPS) server. How can I use openssl s_client to verify that I've done this?
OpenSSL s_client
To check if you have disabled the SSLv3 support, then run the following
openssl s_client -connect example.com:443 -ssl3

which should produce something like
3073927320:error:14094410:SSL routines:SSL3_READ_BYTES:sslv3 alert handshake failure:s3_pkt.c:1258:SSL alert number 40
3073927320:error:1409E0E5:SSL routines:SSL3_WRITE_BYTES:ssl handshake failure:s3_pkt.c:596:

meaning SSLv3 is disabled on the server.  Otherwise the connection will established successfully.
Nmap
Alternatively, you can use nmap to scan server for supported version:
# nmap --script ssl-enum-ciphers example.com
Starting Nmap 6.47 ( http://nmap.org ) at 2014-10-15 03:19 PDT
Nmap scan report for example.com (203.0.113.100)
Host is up (0.090s latency).
rDNS record for 203.0.113.100: edge.example.com
Not shown: 997 filtered ports
PORT    STATE SERVICE
80/tcp  open  http
443/tcp open  https
| ssl-enum-ciphers: 
|   **SSLv3: No supported ciphers found**
|   TLSv1.0: 

