I find Griffintown is a bit overrated. It's major construction all around and the prices are no longer low. So no point in suffering through all that.
If you want to live in up-and-coming neighborhood before it gets spoiled, I'd check out Verdun, St-Henri areas.
Plateau is a great and developed area with history, but not cheap.
So as you can see - it all depends. I think it would be more useful if you posted your budget and other requirements (e.g. public transport or car, amenities, restaurants or costco, etc....)