Q.  When can I play a 1 Tribble?
A.  On your turn, when you are executing orders.  Because your Tribble Side Deck involves drawing cards, not just playing cards, using it will advance you out of your card play phase.

Q.  How many 1 Tribbles can I use each turn?
A.  One.  Since the ability is “once per turn” and Tribbles are not cumulative, you can only use the ability of one 1 Tribble every turn.  (You’ll find this under “once per turn” in the glossary, not under “cumulative”.)  Of course, you can have multiple 1 Tribbles in play to keep the opponent guessing about which one you’ll use.

Q.  Exactly when can I use 1 Tribble?
A.  Players alternate taking actions, with the active player getting the first action on their turn.  So if it’s the opponent’s turn, they get the first action, which is typically a card play,  then you get a chance to use a 1 Tribble to stop someone.  If the opponent skips their  card play and uses their first action to more someone away from a 1 Tribble, then you won’t have a chance to stop that person.  The opponent does not have to “pass” on their card play to move to their executing orders phase. 
(In fact, if the opponent wanted to, the first action on their turn could be to end their turn and draw a card.) Note that “start of turn” abilities are actions.  So if the opponent Processes Ore, you then get a chance to use a 1 Tribble.  However, each player can take as many “start of turn” actions as they want before anyone can take a non-“start of turn” action. 
(This is similar to “just” actions.)  So perhaps the opponent gets to Process Ore AND persona-swap Elim Garak for Elim, THEN you can Tribble them.  But they can’t Process Ore AND play a Q’s Tent without giving you an action in between.

Q.  How do I get rid of small Tribbles?
A.  Small (1 and 10) Tribbles can be carried by like equipment.  So when you move a personnel, they can carry a Tribble with them.  They can keep carrying the Tribble as long as they want, but when they eventually decide to drop it, they are stopped. Let me give you a detailed example: say you have a 1 Tribble on an outpost, and Mendon is present.  Your action is to move Mendon from the Outpost onto the Danube that is docked there, he carries the Tribble with him when he goes.  The Tribble is now “stuck” with Mendon until he decides to drop it.  Mendon is still not stopped at this time, and can still staff that Danube and fly it to another location.  He can also beam down from the Danube, still with the Tribble.  In fact, you have no choice, the Tribble has to keep going with him until you say it’s dropped.  THEN, you say Mendon is going to drop the Tribble, which stops him on the planet.

Q.  Isn’t that a lot of bookkeeping for one Tribble?
A.  Because you can take many actions while carrying Tribbles, you have to let your opponent know when your personnel is finally dropping the Tribble and becomes stopped.  It’s not that confusing if you’re aware of it. 
(If the rules worked the other way, if Tribbles were automatically dropped when moved, it would be very annoying. Mendon would be stopped after boarding the Danube and it would take many turns or hand-offs between many personnel to carry away 1 Tribble.)  Of course, it can often be assumed that if you beam down Mendon and his Tribble and leave them there until end of turn, then he has dropped the Tribble.  But people also assume the opponent knows that ships were docked when they don’t specifically say they were, and we all know what chaos that can cause.  It’s a minor little bookkeeping that will prevent arguments like “no, that Tribble is still stuck with Mendon from last turn.”

Chris 'Rom' Brennan Rom@wnohgb.com