S.E.
Only children can be trying at times. Our son is an only child and he has made it to 19 and we are all still alive and doing fine.
Unfortunately ignoring is a hard thing but sometimes that is what you need to do. Do not give her the satisfaction that she might be winning because let me tell you only children can be very smart. They learn really quick how to get things they want because they know they are the baby, the one and only baby. Our son did not talk til he was over 2 years old. We asked his doctor one day do you think there is something wrong? Should we get him checked out. We thought because he such ear problems when he was small and already had tubes put in there might be some problems. THe doctor did not say much at first be sat and talked to us for a while and the doctor said to us, "The problem is not with Hans (our son) it is with you." We said what?? He said "I took extra time with you today to really check your son out good and talk to you to see how Hans is really doing." He said, "Hans does not need to talk. I he has to do is point or move one way or another and you get him what he wants or do what ever whats with out him saying a word." When we stopped doing everything without even thinking he started to talk and ask for things. ONly children know the score and if they are the only Grandchild, they have more working for their side. They know how to get spoiled.
This is what I would suggest. When she has a fit come down to her level to talk to her. Tell her that you need to tell me in your own way what you want. Eye to eye is best when talking to a little one. Do not show any signs of being upset or made. Be kind but firm. You could even make up some card with the word water or milk and a picture of a bottle, or a picture of her favorite toy what she would like. It also is an exercise in early reading and speaking. You can ask her to say the name of the item. You can also give her bonuses for not whining and using the cards to ask you for something.
If you think she is being naughty and not just moody there is timeout and you can put at 15 month old in timeout.
Well there is a couple ideas, I could go on and on about raising an only child but not enough time. Also you can check out www.focusonthefamily.com and see what they have to say about the only child. Mr. Dobson I know wrote a book about the only child. I am sure he has some advice on that website.
Good Luck