1. It is never going to be inappropriate to send a thank you note, so if you are in doubt as to whether or not to send a thank you note, then send one. Basically, if you aren't sure if you need to send one, then do it, because if someone does not expect a thank you, and receives one anyway, they are going to be extra grateful.
2. Wedding gifts require thank you notes. It is especially important to send a thank you note if the gift was sent from an out of towner. If they were unable to attend your wedding, reception, etc. and sent a gift anyway, then a thank you is a big way of showing them your thanks and keeping yourself in their good graces. Even if you thanked someone for coming to your wedding it is considered appropriate manners to send them a thank you. Of course, they are going to be understanding of the fact that you are busy, so courtesy says you have a year in which to get your thank you notes sent, but most etiquette rules dictate that within 6 months, if not 6 weeks is a better time frame for wedding gifts.
3. Shower gifts tend to be the subject of a lot of debate when it comes to thank you gifts. Because wedding presents are often opened outside of the presence of the giver, a thank you card is your way of letting them know you not only received the gift, but were grateful for it. A shower, is more intimate, and you open the gifts there in front of the guests, so it is not as necessary to send out a thank you to tell them you liked the gift, as you can do it in person. So, some would say that a thank you gift is unnecessary for those who attend your shower, but others would say that any time a gift is given, a thank you card is expected, if not required. So again, judge based on who gave you the gift, what they would expect, and when in doubt, send a thank you card.
There are several other incidences when a thank you card may be appropriate. Basically any time someone lays out some money to get you a gift, or celebrate something with you, or for you, then it is appropriate and polite to send a thank you card to indicate that you received their efforts, are grateful to them, and appreciate their participation.

One note, while etiquette dictates a time frame in which a thank you card should be sent, no one is going to complain about receiving a thank you card later. So, if you never wrote out your wedding thank you notes, then do it now, and get them sent out, even if it has been a few years.