[windows] What's the best way to determine which version of Oracle client I'm running?

you can use the following command in SQL Developer or SQLPLUS in command prompt to find out the Oracle server version number.

select * from v$version;

in my case it gave me the below mentioned info.

Oracle Database 11g Enterprise Edition Release 11.2.0.1.0 - 64bit Production
PL/SQL Release 11.2.0.1.0 - Production
"CORE   11.2.0.1.0  Production"
TNS for 64-bit Windows: Version 11.2.0.1.0 - Production
NLSRTL Version 11.2.0.1.0 - Production