Why the requirement to put everything in one trigger? Makes life complicated, I think...
You can write a trigger on all three actions.
CREATE TRIGGER tr_InsUpdDel ON tableA
FOR INSERT, UPDATE, DELETE
AS
...
Inside the trigger you can check the count of the Inserted and Deleted tables :
Deleted = 0, inserted <> 0 ==> INSERT
Deleted = Inserted ==> UPDATE
Deleted <> 0, inserted = 0 ==> DELETE