I've read that there is a type of luck called "love luck". In the article, it states that love luck either makes it easier to increase friendship with opposite-sex villagers, or that special conversations with opposite-sex villagers are more likely (if you have 'good'love luck). I have yet to witness the effect of this luck, but I would like to see some examples of this "love luck" to see if it's worth the 500 Bells that Katrina requests to have your fortune read.
First off, the type of luck you receive each day is random, and Katrina doesn't "give" you the luck--you have it already and she tells you what kind you have.
And so for "love luck," it should already be easier that day to increase friendship with opposite-gender villagers and to get special conversations with them. You don't need Katrina to unlock your luck, just to clarify what it is for the day. I can attest to "love luck" working as true (or having just several weird coincidences); the number of special conversations I had with opposite-gender villagers with good "love luck" was higher than normal (and definitely more than the ones I had with same-gender villagers).
I think that the relationship that grows is not "love luck" i think its just the time you have spent with the villager. On the other hand, there is a way for you to have a serious love relationship with a villager of an opposite sex.....You have to send them alot of letters with roses and talk to them and if there are hearts on top of his/her head he or she is in love with you or if he or she is lazy he might be in love with food .