How to check what type of lock an UPDATE statement uses? How can I determine if an update statement puts a table lock, page lock, or row lock on a table?
You can see the locks from DMV called sys.dm_tran_locks. The type is of lock can be seen from resource_type column (f. ex. OBJECT, PAGE etc).

More information is available in MSDN 

Also, maybe this query by Jonathan Kehayias is useful:

SELECT dm_tran_locks.request_session_id,
       dm_tran_locks.resource_database_id,
       DB_NAME(dm_tran_locks.resource_database_id) AS dbname,
       CASE
           WHEN resource_type = 'object'
               THEN OBJECT_NAME(dm_tran_locks.resource_associated_entity_id)
           ELSE OBJECT_NAME(partitions.OBJECT_ID)
       END AS ObjectName,
       partitions.index_id,
       indexes.name AS index_name,
       dm_tran_locks.resource_type,
       dm_tran_locks.resource_description,
       dm_tran_locks.resource_associated_entity_id,
       dm_tran_locks.request_mode,
       dm_tran_locks.request_status
FROM sys.dm_tran_locks
LEFT JOIN sys.partitions ON partitions.hobt_id = dm_tran_locks.resource_associated_entity_id
JOIN sys.indexes ON indexes.OBJECT_ID = partitions.OBJECT_ID AND indexes.index_id = partitions.index_id
WHERE resource_associated_entity_id &gt; 0
  AND resource_database_id = DB_ID()
ORDER BY request_session_id, resource_associated_entity_id