Actually, this section depends a lot on your interests, your budget, if you are alone or not, etc. A "must see" sight for someone might not be as interesting for someone else. So I'll try to add different lists depending on various preferences. For the moment, here's a general list.
If you are going to be in Montreal, and only have a few days, you must at least visit these places :
Old Montreal and the old port , it may get crowded in the summer but it's so nice to walk around and look at the sights. There's also a lot of family activities in the old port and food (but usually expensive) restaurants;
Ste-Catherine street downtown
between Square-Victoria (Union street) and Guy street, especially at night or in the afternoon. It's the main commercial street but it's more for walking then for shopping. You can also have access to the underground "city" (if you would like to do even more shopping... and more choice);
The Mont-Royal park and its lookout, the best view of Montreal. If you want the city at your feet, that's the best way to do it;
The Plateau Mont-Royal district, a visit to Montreal is not complete without a least a little walk in that district and a stop at one of the neighborhood restaurants.
St-Laurent boulevard north of Sherbrooke street for its restaurants and bars/clubs;
St-Denis north of Ste-Catherine (lot's of restaurants, boutiques, book stores and coffee shops) in the Quartier Latin district. The other part of St-Denis (north of Roy) street is even more interesting (the plateau Mont-Royal district);
The "islands" (Ile
Ste-Hélène and Ile Notre-Dame), for the La Ronde amusement park, the Casino de Montréal, the bicycle paths and hiking paths, the old fort, the Expo 67 site and the Floralies site, etc.
there are lot's of them so there's a lot of variety, just don't
go to the usual places.
The festivals, almost too numerous
to mention, like the Montreal International Jazz Festival, the
Just for Laughs festival, the International Film Festival (and
the other 14 film festivals held every year), the Francofolies,