It seems to me that you do not want a global variable but, instead, an instance variable. In that case, your declaration:
@interface MainViewController : UIViewController <FlipsideViewControllerDelegate> {
    NSMutableArray *sizedWordList;
}
in the header file is correct. However, in the implementation file, you cannot do the following outside of an instance method (or, if it were indeed a global variable, outside of a class method or a function):
sizedWordList = [[NSMutableArray alloc] init];
It is not legal in Objective-C. The correct place to initialise instance variables is the -init method. Since your class is a subclass of UIViewController, you should override its designated initialiser, -initWithNibName:bundle::
- (id)initWithNibName:(NSString *)nibName bundle:(NSBundle *)nibBundle {
    self = [super initWithNibName:nibName bundle:nibBundle];
    if (self) {
        sizedWordList = [[NSMutableArray alloc] init];
    }
    return self;
}
Your -dealloc method is almost correct — remember that you should always send [super dealloc] at the end of your -dealloc method:
- (void)dealloc
{
    [sizedWordList release];
    [super dealloc];
}
Having done that, you can use the array in any other instance method. For instance,
- (void)logWordList {
    NSLog(@"%@", sizedWordList);
}