Need an Oracle query to get the Oracle Home directory for a particular instance

We are mounting a database to an instance. See the concepts guide:

The instance mounts a database to associate the database with that instance.

One way I've seen it explained (probably here or on SO) is that the instance is the memory structures and processes, and the database is the data on disk. The instance has to have a database associated with it to be able to do anything useful.

First of all, I should thank you for all other steps that you mentioned above. As an addition to your points, I have this one which I've been using so far.

Solaris and Linux

$ pgrep  -lf _pmon_
12586 ora_pmon_ee10204

ORACLE_SID is ee10204

$ pwdx 12586
12586: /u01/oracle/10.2.0.4/ee1/dbs

ORACLE_HOME is /u01/oracle/10.2.0.4/ee1

HPUX

$ ps -ef | grep pmon
ora1024 25611     1  0  Mar 21  ?         0:24 ora_pmon_itanic10

ORACLE_SID is itanic10

$ pfiles 25611 | grep  bin
25611:                  /opt/ora1024/app/db_1/bin/oracle

ORACLE_HOME is /opt/ora1024/app/db_1

AIX

$ ps -ef | grep pmon
ora1024   262314        1   0   Mar 23      -  0:12 ora_pmon_mercury

ORACLE_SID is mercury

$ ls -l /proc/262314/cwd
lr-x------   2 ora1024  dba  0 Mar 23 19:31 cwd -> /data/opt/app/product/10.2.0.4/db_1/dbs/

ORACLE_HOME is /data/opt/app/product/10.2.0.4/db_1

Personally, I haven't seen any environment that tries to confuses the person who will maintain the system. Most of the time, I've always seen the normal /u01/app/oracle for the ORACLE_BASE.