Searching with monsters present--allowed!
After ruminating on the rules for line of sight, including the terms "see" and "visible", it occurs to me for the first time there is a loophole available to Heroes who wish to search a room for traps or secret doors while it still contains one or more monsters. I'll open with the NA Instruction Book, p.15:

I just want to consider the Elf, Wizard, and Orc that isn't visible directly to the right. Now check the relevant search rule for secret doors on p.17, found under How A Hero Searches For Secret Doors:
Likewise, the rule for searching for traps on p.18 states the same under How A Hero Searches For Traps:
It is also worth noting that a search for treasure is worded and thus handled differently. This search works as you would expect (p.16, How a Hero Searches For Treasure):
Finally, for the EU players, the relevant rule on p.12 under Searching:
Searches are more restricted in rooms, as none are possible while a monster is present. This is similar to searches for treasure in the NA rules. However, since treasure may be searched for in corridors using EU rules, it may be done with monsters present as long as they are obstructed (not visible). Searches for traps and secret doors may also be allowed in corridors with monsters that are obstructed.
I haven't heard of anyone using the rules to their advantage in this way before, but it seems like it is allowed.

I just want to consider the Elf, Wizard, and Orc that isn't visible directly to the right. Now check the relevant search rule for secret doors on p.17, found under How A Hero Searches For Secret Doors:
- * As a Hero , you can only search for
secret doors if there are no monsters
visible to you.
Likewise, the rule for searching for traps on p.18 states the same under How A Hero Searches For Traps:
- * As a Hero , you can only search for
traps if there are no monsters visible
to you.
It is also worth noting that a search for treasure is worded and thus handled differently. This search works as you would expect (p.16, How a Hero Searches For Treasure):
- * As a Hero, you may search a room
for treasure only if the room is
uninihabited by monsters. [edit in red]
Finally, for the EU players, the relevant rule on p.12 under Searching:
- ...Characters may not search if they are next to a monster or if there is a monster in the same room or visible in a passage....
Searches are more restricted in rooms, as none are possible while a monster is present. This is similar to searches for treasure in the NA rules. However, since treasure may be searched for in corridors using EU rules, it may be done with monsters present as long as they are obstructed (not visible). Searches for traps and secret doors may also be allowed in corridors with monsters that are obstructed.
I haven't heard of anyone using the rules to their advantage in this way before, but it seems like it is allowed.