Database triggers are well-matched ways of monitoring your database and logging events you'd like know about. It's possible to capture errors occurred and even notify yourself for exceptional situations. An example of these benefits is what I'll briefly present here. Below you'll find a demonstration of a trigger which logs errors and sends a mail when out of space error occurs.

First of all, start with creating the table to log into:

CREATE TABLE SERVERERROR_LOG
(
ERROR_TIME TIMESTAMP,
DB_USER VARCHAR2(30),
OS_USER VARCHAR2(30),
USER_HOST VARCHAR2(64),
CLIENT_PROGRAM VARCHAR2(48),
SQLID VARCHAR2(13),
ERROR_STACK VARCHAR2(2000)
);

Table keeps records of SQL running (actually ID of the SQL to save space, since full SQL text can be queried) and the error it caused beside client information. Here is the trigger:

CREATE OR REPLACE TRIGGER log_server_errors
AFTER SERVERERROR ON DATABASE
DECLARE
osuser VARCHAR2(30);
machine VARCHAR2(64);
prog VARCHAR2(48);
sqlid VARCHAR2(13);
err_type VARCHAR2(19);
obj_type VARCHAR2(19);
owner VARCHAR2(30);
ts VARCHAR2(30);
obj VARCHAR2(30);
sub_obj VARCHAR2(19);
BEGIN
SELECT osuser, machine, program, sql_id
INTO osuser, machine, prog, sqlid
FROM gv$session
WHERE audsid = userenv('sessionid') and inst_id = userenv('instance') and status = 'ACTIVE';

INSERT INTO servererror_log VALUES
(systimestamp, sys.login_user, osuser, machine, prog, sqlid, dbms_utility.format_error_stack);
COMMIT;

IF (SPACE_ERROR_INFO(err_type,obj_type,owner,ts,obj,sub_obj) = TRUE) THEN
UTL_MAIL.SEND( sender => 'oradb@xyz.com',
recipients => 'dba@xyz.com',
subject => 'Insufficient Tablespace Size',
message => err_type || ' at tablespace ' || ts
);
END IF;
END log_server_errors;

Trigger queries gv$session view to gather client information and inserts to log table. Then, if error is a out of space error, trigger sends an e-mail using UTL_MAIL package. This's a pretty straightforward trigger you can use and improve according to your needs.