I would need to see the context in which you are trying to use just "am" but I can't think of any case in which you can just say "am". You have to say "I am happy." / "I'm happy." or "I am coming." / "I'm coming." The sentences "Am happy." and "Am coming." are incorrect. Sometimes in informal American English, we do drop subjects and objects like you do in Spanish. For instance, if you are on the phone with someone and you suddenly go quiet and then come back after 5 minutes, you can say: "Sorry. Had to get a drink." instead of "Sorry. I had to get a drink." This is informal spoken English though and you should only use it with close friends and family, never in a formal setting like an interview. :)

As Shawn says, in general we can't leave out the subject of a sentence in the same way as you can in some Latin languages. Our verb forms don't vary much, so you need to hear the subject to understand the sentence. With some verbs and in some phrases, though, you can drop the subject, in informal speech. 'Got to run! Bye!' is an example of a informal phrase without a subject. I wouldn't recommend you try this, however. Native speakers know instinctively when you can and can't drop the subject, and - in most cases - you can't. As for 'am' instead of 'I'm' , no, we never say that. This is probably because it's just as quick and easier to say the full form. The only time that 'am' is ever used without the 'I' is when we are writing informal notes, messages, texts and so on. You might write a note to your friend or family saying something like 'Am in town. Back later'. This 'telegram' style of writing is quite common in casual messages.

No you can't say "am" instead of "I am", but you can say "I'm". For example, instead of saying "I am going to the store" you can say "I'm going to the store".