How to fix a Macbook Pro trackpad where the mouse pointer is randomly moving? The trackpad on my Macbook Pro just started acting oddly. It's randomly clicking (which might cause me to switch programs), right-clicking and even once my screen even showed the swiping animation as if I was trying to switch to a different desktop.    

Part of me fears that this is some sort of joke hacking attempt (I know of a USB device you plug into someones computer and it randomly moves their mouse and types on their keyboard), but there is nothing plugged into my machine and I just turned off the Wi-Fi and watched as this web page tried to close, the mouse right clicked twice, highlighted a word and clicked "Paste and Match Style" in the Chrome right-click menu.
Also as I've been typing (with my Wi-Fi turned off) the mouse has randomly been clicking inside this question and changing where I am typing.  

I just plugged in a USB mouse which seems to work fine but the trackpad is now nearly useless. I can't even move the mouse cursor more than a few centimeters with it.

Is this a common issue?
EDIT
I think it is dying actually. I can now click but not move at all with the trackpad, while a regular mouse works fine.
EDIT 2
And now it seems to be working again. I turned on the option for "Ignore Trackpad when Mouse is plugged in" under Universal Access. I cleaned the trackpad with rubbing alcohol, I whined and complained for a few minutes to my dog, turned off the "Ignore Trackpad" setting, unplugged the USB mouse, and the Trackpad appears to be working mostly normally.
If I run my finger across it at the top or bottom portion of the mouse, it works, but there is a line horizontally across it (almost exactly where a physical trackpad button would end on the old trackpads) that is "dead".  The mouse stops moving, or moves sluggishly when hitting that spot and that "spot" goes across the whole trackpad.
This was driving me crazy, too. It just started happening yesterday, but was dangerous as the cursor kept selecting everything--making selection rectangles especially on the desktop, opening programs I didn't want to open on the dock, and generally jumping around like a drunk rabbit. 

Searched forums, read about others saying this happened to them after updating their Mac software, etc., etc. Got so frustrated I literally slapped the trackpad with my open palm. And what do you know -- that fixed it! Now my trackpad is working perfectly again. Must have been something pinching or pressing against the underside of the trackpad. 

Don't know if you're all going to believe me 'cause it sounds ridiculous, and if you do believe me I don't know if you're willing to do it. But I swear on my soul it's true and I'm posting this solution for all to see because I want to help.