Need to list all triggers in SQL Server database with table name and table's schema

Here's one way:

SELECT 
     sysobjects.name AS trigger_name 
    ,USER_NAME(sysobjects.uid) AS trigger_owner 
    ,s.name AS table_schema 
    ,OBJECT_NAME(parent_obj) AS table_name 
    ,OBJECTPROPERTY( id, 'ExecIsUpdateTrigger') AS isupdate 
    ,OBJECTPROPERTY( id, 'ExecIsDeleteTrigger') AS isdelete 
    ,OBJECTPROPERTY( id, 'ExecIsInsertTrigger') AS isinsert 
    ,OBJECTPROPERTY( id, 'ExecIsAfterTrigger') AS isafter 
    ,OBJECTPROPERTY( id, 'ExecIsInsteadOfTrigger') AS isinsteadof 
    ,OBJECTPROPERTY(id, 'ExecIsTriggerDisabled') AS [disabled] 
FROM sysobjects 

INNER JOIN sysusers 
    ON sysobjects.uid = sysusers.uid 

INNER JOIN sys.tables t 
    ON sysobjects.parent_obj = t.object_id 

INNER JOIN sys.schemas s 
    ON t.schema_id = s.schema_id 

WHERE sysobjects.type = 'TR' 

EDIT: Commented out join to sysusers for query to work on AdventureWorks2008.

SELECT 
     sysobjects.name AS trigger_name 
    ,USER_NAME(sysobjects.uid) AS trigger_owner 
    ,s.name AS table_schema 
    ,OBJECT_NAME(parent_obj) AS table_name 
    ,OBJECTPROPERTY( id, 'ExecIsUpdateTrigger') AS isupdate 
    ,OBJECTPROPERTY( id, 'ExecIsDeleteTrigger') AS isdelete 
    ,OBJECTPROPERTY( id, 'ExecIsInsertTrigger') AS isinsert 
    ,OBJECTPROPERTY( id, 'ExecIsAfterTrigger') AS isafter 
    ,OBJECTPROPERTY( id, 'ExecIsInsteadOfTrigger') AS isinsteadof 
    ,OBJECTPROPERTY(id, 'ExecIsTriggerDisabled') AS [disabled] 
FROM sysobjects 
/*
INNER JOIN sysusers 
    ON sysobjects.uid = sysusers.uid 
*/  
INNER JOIN sys.tables t 
    ON sysobjects.parent_obj = t.object_id 

INNER JOIN sys.schemas s 
    ON t.schema_id = s.schema_id 
WHERE sysobjects.type = 'TR' 

EDIT 2: For SQL 2000

SELECT 
     o.name AS trigger_name 
    ,'x' AS trigger_owner 
    /*USER_NAME(o.uid)*/ 
    ,s.name AS table_schema 
    ,OBJECT_NAME(o.parent_obj) AS table_name 
    ,OBJECTPROPERTY(o.id, 'ExecIsUpdateTrigger') AS isupdate 
    ,OBJECTPROPERTY(o.id, 'ExecIsDeleteTrigger') AS isdelete 
    ,OBJECTPROPERTY(o.id, 'ExecIsInsertTrigger') AS isinsert 
    ,OBJECTPROPERTY(o.id, 'ExecIsAfterTrigger') AS isafter 
    ,OBJECTPROPERTY(o.id, 'ExecIsInsteadOfTrigger') AS isinsteadof 
    ,OBJECTPROPERTY(o.id, 'ExecIsTriggerDisabled') AS [disabled] 
FROM sysobjects AS o 
/*
INNER JOIN sysusers 
    ON sysobjects.uid = sysusers.uid 
*/  
INNER JOIN sysobjects AS o2 
    ON o.parent_obj = o2.id 

INNER JOIN sysusers AS s 
    ON o2.uid = s.uid 

WHERE o.type = 'TR'

Here you go.

    SELECT
    [so].[name] AS [trigger_name],
    USER_NAME([so].[uid]) AS [trigger_owner],
    USER_NAME([so2].[uid]) AS [table_schema],
    OBJECT_NAME([so].[parent_obj]) AS [table_name],
    OBJECTPROPERTY( [so].[id], 'ExecIsUpdateTrigger') AS [isupdate],
    OBJECTPROPERTY( [so].[id], 'ExecIsDeleteTrigger') AS [isdelete],
    OBJECTPROPERTY( [so].[id], 'ExecIsInsertTrigger') AS [isinsert],
    OBJECTPROPERTY( [so].[id], 'ExecIsAfterTrigger') AS [isafter],
    OBJECTPROPERTY( [so].[id], 'ExecIsInsteadOfTrigger') AS [isinsteadof],
    OBJECTPROPERTY([so].[id], 'ExecIsTriggerDisabled') AS [disabled] 
FROM sysobjects AS [so]
INNER JOIN sysobjects AS so2 ON so.parent_obj = so2.Id
WHERE [so].[type] = 'TR'

A couple of things here...

Also I see that you were attempting to pull the parent tables schema information, I believe in order to do so you would also need to join the sysobjects table on itself so that you can correctly get the schema information for the parent table. the query above does this. Also the sysusers table wasn't needed in the results so that Join has been removed.

tested with SQL 2000, SQL 2005, and SQL 2008 R2

I had the same task recently and I used the following for sql server 2012 db. Use management studio and connect to the database you want to search. Then execute the following script.

Select 
[tgr].[name] as [trigger name], 
[tbl].[name] as [table name]

from sysobjects tgr 

join sysobjects tbl
on tgr.parent_obj = tbl.id

WHERE tgr.xtype = 'TR'