
To start off, I am going to include the examples that Justin gave in the stem thread. Please jump in with further examples!
A. Offering a pin play (or drop)
1. You hold T9x and Dummy on your right holds AJ8xx. Play the T or 9. (If Declarer holds Qxx (partner Kx) then this gives the losing option of playing for T9 doubleton).
2. You hold JTx. Declarer has a suit of AQ9xxxxx opposite a void (either way) giving partner Kx. Drop the J or T under the ace. (This gives Declarer the losing option of playing for JT doubleton).
3. You hold JTx. Declarer holds x on your left and KQ98xxx on your right. When Declarer plays up to the K or Q you must drop an honour. (This give Declarer a guess as to whether we started with JTx opp Ax or JT opp Axx.)
4. You hold Txx. Declarer has Ax on your left and KJ98x on your right. Drop the T under the ace to give Declarer the chance of playing you for QT bare. (Otherwise Declarer will have to play partner for Qxx to pick up the suit.)
B. Offering a two-way finesse
1. You hold J9xx. Declarer holds AK8x on your right and QTxx on your left. Drop the 9 under the ace to set up a two-way finesse for the jack.
2. As above but AKTx on your left and Q8xx on your right.
3. You hold KJx. Declarer has Axx on your left and QT98 on your right and plays up to the T on the first round. Win the king to allow Declarer to finesse partner for the jack. (If you win the J Declarer has no choice but to finesse you for the K.)
C. Creating a losing finesse position
1. You hold KT. Declarer has AQxxx on your left and J9x on your right. When Declarer leads low to the Q you must play the king. (This gives Declarer the losing option of finessing the 9, playing your partner for Txxx.)
2. You hold QT. Declarer has AKxxx on your left and J9x on your right. When Declarer leads low to the AK you must play the queen. (This gives Declarer the losing option of finessing the 9, playing your partner for Txxx.)
3. You hold JTx with AK9xxx on your right. When Declarer plays the queen from your left, drop an honour. (This gives Declarer the option of playing us for a singleton and taking the finesse.)
4. You hold KT8x. Declarer has Q9xx left and AJ7x right. When Declarer finesses the J you can try dropping the 8 under it. (This gives Declarer the possibility of trying to run the Q next, potentially creating a second trick in the suit.
5. You hold QJ9x. Declarer has KT876 on your left and Axx to the right. When Declarer plays a small card on your left, play the 9. (This gives Declarer the option of going up with the K to try running the T.)
6. You hold KJ bare with AQxxx on our left and T8x on our right (partner 9xx). Declarer leads the x from our right. Playing the K means Declarer will most likely play to the T allowing our J to win a trick.
7. You hold Jx with AKTx on your left and declarer on your right has shown 5. When declarer plays up to the AKT it cannot hurt to drop the jack. If partner holds Qx then declarer might decide you split honours and finesse on the second round.
Not quite falsecarding but related is the concept of playing the card you are known to hold. Some examples of this (there are probably too many to include them all):
D: Playing the known card
1. You hold T8x with AJ9xx on your left. Declarer finesses the 9 drawing an honour from partner. When Declarer now plays towards the AJxx you must play the T. (Declarer might now play you for HTx.)
2. You hold QTx. Declarer has AJx on your left with K9xx on your right and begins by finessing the J. When Declarer play the A on the second round you must play the Q. (Declarer can play you for Qx bare.)
E: Non-mandatory falsecards
1. Winning a trick with the ace instead of the queen from AQx(x) to encourage declarer to continue the suit when you can see another suit is breaking well for her/him.
2. Dropping an honour to encourage declarer to use up an important entry. As an example, you hold JTxx with K98x on your left and AQ76 on your right. If you drop the jack, declarer will almost certainly cross to the king next.
In addition, if there is interest we could also add standard positions where you must randomise your cards. I suspect that might be a large topic though and perhaps a separate thread for it would be better - opinions gladly accepted on this.