I really like Core Secrets and Pilates for all of the areas you listed. Core secrets is the one with the big ball. Boy oh boy, the one that really sticks out is hamstrings with core secrets. There is this one exercise that if you lay on your back with your feet on the ball and lift your butt equal with the ball...you will not be able to walk the next day. It wears my leg muscles out.

I really don't like doing ONE exercise for a muscle because I don't think it properly develops the muscle from all angles and through all ranges of motion. Also, everyone's body is different- you have a unique history which may make some exercises better or worse for you. For example. I love to do a lot of weighted exercises standing one-legged on the flat side of a bosu ball, but a less experienced lifter or someone with less well developed balance might kill themselves doing that. Also, if your primary goal is muscle hypertrophy, it's not very useful.

However, I'll play your game:

Shoulders- overhead press with dumbbells, done standing
Biceps - dumbbell curls with a twist, done standing on a bosu ball
Triceps - cable rope overhead extensions
Chest - OK, I can't give ONE: dumbbell presses on a stability ball, pushups, dumbbell flyes, cable flyes, more pushups.
Back- Which part of your back? Pullups and dumbbell one arm rows are a minimum.
Abs - Hanging "knee ups" or "leg ups" if you are strong enough
Obliques Hanging obliques
Lower Abs - There isn't any such thing. It's one muscle.
Butt - Deep squats and lunges
Quadriceps - Deep squats and lunges
Hamstrings - one-legged romanian deadlifts
Calves - do as much on a bosu ball as you can and you don't need to directly work your calves

So what is your program?

Mel

Unfortunately, I never know what they're really called in the training world:

Biceps - reverse skull crusher
Triceps - skull crusher
Chest - the rowing thingy
Obliques - side crunches using a disk (anyone use this? it's great!)
Lower Abs - toe drops
Quadriceps - squats rolling down on a ball
Hamstrings - the same one Stacy mentions
Calves - calf raise

Mel, about the abs. When I do regular crunches, I get an effect more near my belly button. When I do toe drops, I can feel it more near the pubic area (i.e., "lower abs"). My trainer gave me more exercises that seem to work on the "lower" abs, even though it's all one muscle, because she said that area was weaker.

See one reason I'd be scared of Mel as my personal trainer... Look at her list... Ok not really

I can't say the best because well the best is to challenge your muscles with different exercises but I'll say what I like

Shoulders- I like overhead presses but I like to do them seated.
Biceps - hammer curls.
Triceps - bent over tricep kick backs
Chest - dumbbell flys lying on a bench
Back- assisted pullups (one day I hope to do unassisted pullups but I need to lose weight and increase strength before that day comes)
Abs - Planks!
Obliques - twisting crunches
Lower Abs - reverse crunches
Butt - Lunges! Although I have a new favorite which I got from my Push DVD. It is a bridged kick up. Basically you do a bridge, then straigten one leg and kick up. My booty hurt for 2 days due to that one.
Quadriceps - I have to say I have always loved the overhead lying leg press where you push weight up while you are lying down.
Hamstrings - I like dead lifts, I may have to try them with one leg now and see what happens.
Calves - donkey raises

Shoulders - overhead press and side raises
Biceps - tend to get a good workout on the lat pulldown and the seated rower so I don't target them specifically otherwise I'd never be able to punch the next day!
Triceps - pushdown with the lat pulldown machine
Chest - bench press
Back - lateral pulldown with drop sets
Abs - inverted hanging abs or knee-ups or high crunches with a partner
Obliques - oblique crunches with plates and/or on a stability ball
Butt - deadlifts
Quads - running Ok that's not a weight exercise, erm... One legged squats with or without weights depending on the running and kickboxing schedule!
Hammies - er... leg curls I suppose, but usually kickboxing
Calves - don't work my calves at all - they are already BULGING and I want them to go down! A good one is weighted calf lifts though - I used to rock at those when I did them