This script will list the location of the last Full, Differential and Tlog (if exists)

I use three CTE and at the end I use a UNION all

the last two select statements are joins so it displays NULL if either Diff or Tlog do not exist

 /*
DESCRIPTION    List location of  all backup for a single database.
				Includes if a Tlog or Dif exists or not
CONFIGURATION    

                          Set @LastXday to limit the number of days to search for

                          Set @dbname for the database

 
Compatibility list:
MSSQL2005
MSSQL2008
MSSQL2012
MSSQL2014

 */ 
declare @dbname varchar(500)
declare @LastXdays nvarchar(2)
set @dbname = db_name()
--SET @dbname = 'ELMO'
set @LastXdays = '30'
 
;with fullbk as (  
SELECT
	top 1
	 s.database_name
	, db_id(s.database_name) 'dbid'
	,s.recovery_model
    ,m.physical_device_name
    ,cast(s.backup_size/1000000 as varchar(24))+' '+'MB' as bkSize
    ,DATEDIFF(second,s.backup_start_date , s.backup_finish_date) 'Seconds_TimeTaken'
    ,DATEDIFF(dd,(s.[backup_finish_date]),GETDATE())  AS [age_in_days]
    ,s.backup_start_date
    ,'full' 'bk_type'
	,ROW_NUMBER() over (partition by s.database_name order by s.database_name, s.backup_start_date desc) 'number'
FROM msdb.dbo.backupset s
inner join msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id
WHERE
    s.database_name like @dbname
    and DATEDIFF(dd,(s.[backup_finish_date]),GETDATE()) < @LastXdays
	and s.[type] = 'D'
ORDER BY  s.backup_finish_date desc
)
, Diffbk as (  
SELECT
	top 1
	 s.database_name
	, db_id(s.database_name) 'dbid'
	,s.recovery_model
    ,m.physical_device_name
    ,cast(s.backup_size/1000000 as varchar(24))+' '+'MB' as bkSize
    ,DATEDIFF(second,s.backup_start_date , s.backup_finish_date) 'Seconds_TimeTaken'
    ,DATEDIFF(dd,(s.[backup_finish_date]),GETDATE())  AS [age_in_days]
    ,s.backup_start_date
    ,'Diff' 'bk_type'
	,ROW_NUMBER() over (partition by s.database_name order by s.database_name, s.backup_start_date desc) 'number'
FROM msdb.dbo.backupset s
inner join msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id
WHERE
    s.database_name like @dbname
    and DATEDIFF(dd,(s.[backup_finish_date]),GETDATE()) < @LastXdays
	and s.[type] = 'I'
ORDER BY  s.backup_finish_date desc
)
, TlogBK as (
SELECT
top 1
	 s.database_name
	, db_id(s.database_name) 'dbid'
	,s.recovery_model
    ,m.physical_device_name
	,cast(s.backup_size/1000000 as varchar(24))+' '+'MB' as bkSize
    ,DATEDIFF(second,s.backup_start_date , s.backup_finish_date) 'Seconds_TimeTaken'
    ,DATEDIFF(dd,(s.[backup_finish_date]),GETDATE())  AS [age_in_days]
    ,s.backup_start_date
    ,'Tlog' 'bk_type'
	,ROW_NUMBER() over (partition by s.database_name order by s.database_name, s.backup_start_date desc) 'number'
FROM msdb.dbo.backupset s
inner join msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id
WHERE
    s.database_name like @dbname
    and DATEDIFF(dd,(s.[backup_finish_date]),GETDATE()) < @LastXdays
	and s.[type] = 'L'
ORDER BY  s.backup_finish_date desc
)

select a.database_name, a.recovery_model, a.bk_type, a.physical_device_name,a.bk_type,a.Seconds_TimeTaken,a.age_in_days, a.backup_start_date
from fullbk a
where 1=1
and a.number  = 1
UNION ALL
select c.database_name, c.recovery_model, 'Diff', 
coalesce(b.physical_device_name,'') as 'physical_device_name',
b.bk_type,
b.seconds_TimeTaken,
b.age_in_days,
b.backup_start_date

from Diffbk b right join fullbk c on b.dbid = c.dbid
where 1=1
and c.number  = 1		 

UNION ALL
select c.database_name, c.recovery_model, 'tlog', 
case c.recovery_model
when 'simple' then ''
else b.physical_device_name end as 'physical_device_name',
b.bk_type,
b.seconds_TimeTaken,
b.age_in_days,
b.backup_start_date

from Tlogbk b right join fullbk c on b.dbid = c.dbid
where 1=1
and c.number  = 1