Many events just randomly occur in the game. They don't show up on the schedule or on the Bulliten Board, so you never know when they are going to happen. A few of the events listed here are from later in the game, so don't read unless you don't mind knowing what happens.
If you plant a lot of Melon seeds (around 12 bags will do) you may get this event. When your melons are ready for harvest, Ken may come by in the morning and ask if you would be willing to give him 20 of them. If you tell him yes, he'll ask you to put 20 melons in the shipping bin by 5pm and he'll take care of the rest. He also requests that you don't put anything else in the bin. If you put in 20 melons into the bin like he asks, he'll come back at 5pm and give you 5000G. If you short him and only put in 19 or even less, he'll still pay you but only 150G per melon (which is more then the Buyer pays anyway). If you put MORE then 20 melons in the bin, or you also add in your eggs or milk or other items, you'll only get paid the 5000G for the 20 melons.
If you buy a cow or a sheep, make sure you take care of it! If you don't feed it your animal will get sick. Sick animals have green faces and don't look too healthy. If you don't give it cow or sheep medicine and nurse it back to health it will die a few days later. You get to see the graveyard and have a ceramony with Cain and Chet. Chet will then scold you for not taking care of your sick animal. If you kill more then one animal, he really gets pissed at you. You also loose the whole day by being at the ceramony. Don't forget to feed the surviving animals before you go to bed!
Hot tubbing with Daisy
One day when you go to town you may notice that Daisy's shop is closed even though it's not a Sunday. If you run up into the mountains she'll be there by the Hot Springs. She asks if you want to join her for a bit. If you agree she'll go into the Hot Springs and the two of you can just chill out and relax. She'll stay there all day long too. If you leave the Mountains and come back, she'll be standing on top of the water and she'll give Will's "You should take Sundays off" speach even though it may not be Sunday. Quite strange...
Mary may pay you a visit one morning and asks you to meet with her up in the Mountain. When you go to visit her, she'll be up by the Hot Springs. After a short conversation with you she'll get the idea to write secret messages and bury them in the ground. You can choose from "I want to build a great farm!", "I love Mary!!", "a secret message", "I'm hungry...", a bunch of paw prints, "Harvest Moon 2!", and "I want to play the next version!". Pick one of the selections and choose to Write it. She'll write her message too and then tell you that she is going to bury it right there. She wants you to return to that spot in 3 years (!!!) and not to forget. If when she asks you to meet her or write the message you say "no" enough times, she gets mad at you says she hates you.
Chocola wants you to meet her in the Mountain to hunt for Mushrooms. She'll be right by the entrance when you go to see her. She wants you to go around and hunt for Mushrooms. Pick up the three Mushrooms that are scattered on the ground and give them to her. After the third one talk to her twice and she'll thank you. She also says if you stop by the Restaurant she'll make you Mushroom Rice. A week later there will be a message on the Bulliten Board from Chocola. For the next 10 days she'll sell Mushroom Rice at the Restaurant.
Bill will come to you and tell you that there's been a report of a ghost! Head to the Town Square. Ken and the Mayor will be the only ones there. The Mayor said that he put a notice about the ghost on the Bulliten Board. The notice is just a warning to watch out for the ghost. Talk to Ken and he says he really got scared by the ghost. He then leaves to return to his shop. Talk to the Mayor again and he'll ask you to go see Cain. Cain wants you to ask Ken the details of his story. Go to Ken's shop and ask him. He'll say specific keywords you want to make note of. When you return to the Church, Cain wants you to retell the story like Ken said it. You'll then discover what really went on that night.
Chet and Rosie's Wedding
This isn't exactly a "random" event, but it's not listed on the schedule at first. On the 20th of Spring, third year, Chet comes by and invites you to his wedding. On the 1st of Summer, head over to the Church and watch the festivities. After the wedding Rosie leaves Chocola in charge of the Restaurant.
Make sure you finish your farm chores before heading to town; once this event is finished it automatically becomes night time. In the third year of your farm, head to town on the first Sunday that it's not snowing. Go into the Town Square and you'll almost get hit by a snowball! Bill and Daisy are there throwing snowballs around. Daisy invites you to join them. Will and Chocola will show up at that point. This event is just something fun to do - pick who you want to throw a snowball at and they'll say a little speach. Once the event is done you get a picture of you at the snowball fight in Mary's picturebook.
If you choose to continue playing after you get your farm evaluated another event will occur. One morning you will wake up and not hear anything! If you want to repair your hearing, go to the Clinic and the Doctor will give you some medicine that will fix it the next day. If you dont go to the Clinic and remain deaf for a few days, you'll eventually stay in bed one day and your hearing will return the next moring.