How to know installed Oracle Client is 32 bit or 64 bit?

In Linux:

1) find where is sqlplus located,

[oracle@LINUX db_1]$ `which sqlplus`
/app/oracle/product/11.2.0/db_1/bin/sqlplus

2) Determine the file type,

[oracle@LINUX db_1]$ file /app/oracle/product/11.2.0/db_1/bin/sqlplus

/app/oracle/product/11.2.0/db_1/bin/sqlplus: ELF 64-bit LSB executable, x86-64, version 1 (SYSV), dynamically linked (uses shared libs). For GNU/Linux 2.6.18, not stripped.

Go to %ORACLE_HOME%\inventory\ContentsXML folder and open comps.xml file

Look for <DEP_LIST> on ~second screen.
If following lines have

PLAT="NT_AMD64" then this Oracle Home is 64 bit.

PLAT="NT_X86" then - 32 bit.

You may have both 32-bit and 64-bit Oracle Homes installed.

A simple way to find this out in Windows is to run SQLPlus from your Oracle homes's bin directory and then check Task Manager. If it is a 32-bit version of SQLPlus, you'll see a process on the Processes tab that looks like this:

sqlplus.exe *32

If it is 64-bit, the process will look like this:

sqlplus.exe

One thing that was super easy and worked well for me was doing a TNSPing from a cmd prompt:

TNS Ping Utility for 32-bit Windows: Version 11.2.0.3.0 - Production on 13-MAR-2015 16:35:32