Yeah, we don't really have any of the basic stuff together in one place.
Lapels -- peak and shawl are considered traditional or "correct" for tuxedos (generally called "dinner jackets" or DJs here). The notch lapel, while very common lately, is a carryover from business suits and few tailored clothing enthusiasts would seek out one.
Single-breasted vs. double-breasted applies to any kind of jacket. All the jackets above are single-breasted. The jacket below is double-breasted as characterized by the "overlap" on the closure of the jacket. Double-breasted jackets should almost always be peak lapel.
Cummerbunds and vests should always cover the waistband of your trousers as that should never be visible in a DJ. If you're wearing a DB jacket, it should remain buttoned covering your waistband so neither a cummerbund nor a vest is required. If you choose a vest, the low-buttoning vests are more traditional than the brightly-colored, high-buttoning monstrosities you'll find in any cheap rental place these days.
Shirts -- typically a turn-down collar is worn with a DJ. Some people prefer wing collars, but those were traditionally worn with white tie/tails rather than a DJ. The front of the shirt can be either pleated or pique. The shirt will have double cuffs requiring cuff links for closure.
Wing collar, pleated front:
Wing collar, pique front:
Turn-down (regular) collar, pleated front:
Shoes should be black patent or highly-polished calf black balmorals. Opera pumps are also correct.
Patent balmorals (altho the captoe is technically incorrect on these):
Do get a white linen or cotton pocket square for your breast pocket like the one on the notch lapel jacket above.
Which part should match your date? Probably none. You might consider a colored pocket square that coordinates with your date if you're really set on the idea. The general concept of semiformal wear for men is that it's very uniform and provides a canvas against which the women look good. That appears to be somewhat lost on most men these days. Try to set yourself apart from the unwashed masses by wearing a good quality, properly fitted traditional DJ rather than some 5-button junk with a neon green vest or whatever they're pushing at the local rent-a-tux.