How to check if your Oracle client is 32 bit or 64 bit?

SOLUTION 1:
If logged into SQL Plus, the banner will tell you 64-bit if the 64-bit version is installed. If it does not specify 64-bit then it is 32-bit (even though it does not explicitly say so).

SOLUTION 2:
You can specifically query the info by running the following:
SELECT * FROM V$VERSION;

The banner will tell you 64-bit if the 64-bit version is installed. If it does not specify 64-bit then it is 32-bit.
SOLUTION 3:
Run the query to check the following:
select distinct address from v$sql where rownum<2;
If the address returned is 16 characters long, it is 64 bit.
If it is 32 bit you will get an 8 character address.

SOLUTION 4:
If the two directories $ORACLE_HOME/lib32 and $ORACLE_HOME/lib are existing then it is 64 bit. If there is only an ORACLE_HOME/lib directory then it is 32 bit client.
*Headsup: In newer versions of the client, the library is not included and this directory may not exist.