There are days I set out for a trail ride and rather not carry anything in terms of tools or other important kit, but the reality is you need to be self sufficient on the trail and even the road. Over the last 20 years of riding a bike I have developed multiple kits that are dependent on where I'm riding, who I'm riding with and the duration I'll be spending outside mashing pedals. Every one has brand loyalty to some extent, so to avoid those pitfalls I'll leave out brands. Here are the kits I ride with. Feel free to share what you carry out on the road or in the backcountry so others can learn or even make adjustments to existing kits.
First, mountain bike rides. Depending on distance, my pack varies quite a bit. Also, if I'm going out with some one who is new to biking, I might add additional items to keep things enjoyable if some thing malfunctions.
Short MTB Rides
These are rides that are about 1-2 hours and I can walk out to my car if some thing breaks on the bike.
The smaller the items the better. I'll either wear them in jersey pockets (not on the spine, usually off to the left or right) or tape the whole kit some where on the bike out of the way. There are some companies who make small hideaway kits that fit into bottle cages and there is even clothing with extra pockets to stash gear.
Long MTB Rides
These are rides that have long miles or go deep into the woods or mountains. Usually there's another person on these longer rides so depending on their skill level I might add more to the kit.
You could obviously add more to this but generally this covers you for those 20 mile, 4-6 hour rides. Weather is a factor, for example when I was in Moab during the summer heat (yeah, I know, stupid) I packed a small sunscreen bottle and extra water with hydration powder mixed in to battle dehydration out in the desert.
I ride bikes with the kids often in the local park. The pace is more casual, with frequent stops to explore various distractions (SQUIRREL!) on the bike path. Typically these items are carried in a pack with some of it packed into the basket of my wife's bike.
Now, I don't carry tubes or hand pump cause the rides we do tend to be walking distance to the car if some one gets a flat. Once the kids are old enough to go longer distances I'll add tubes and hand pumps. Most of the family ride kit is based on quick fixes and enjoying the stops (picnic lunches, etc).
Lets get out there and ride and most of all have fun. Share with us what you like to carry and how you carry it.