Super Mario Bros./Warpless Strategy

See andrewg's WR video for the general strategy. This only goes into detail on the non-obvious things going on.
General note: Using the fire power is faster since killing Bowser skips the bridge collapse animation.

Level by level explanations

World 1-1

When you go into the underground section, start holding "right" when Mario has fallen about half the way down and jump as soon as you're able, trying to land on the very edge of the bricks in the middle. The farther to the left you land, the sooner you start accelerating to full speed. When the pipe transition happens, you should have 380 on the timer.

Later just hit the flagpole at the very top. If you have 370 remaining on the timer, the level is flawless. If you have 369 remaining on the timer, it can also be flawless, but it's very likely you missed the framerule.

World 1-2

The first difficult part here is jumping over the Koopa with the Goomba directly behind it. The best way to do it is to run at full speed and do a jump where you hold A for 1 to 3 frames and then immediately jump again.

TODO: 1-3 through 3-4

World 4-1

This level is trivial. Jump at the top of the flagpole and you have a perfect time.

TODO: 4-2 through 7-4

World 8-1

Even if you do not slow down at all in this level, you may lose a framerule. To get a perfect time in 8-1, you need to have hit the star block around the middle of this level and stomp (kill with fire is okay) no enemies other than the 2 Koopas (explained below).
You can make the part with the 3 Goombas in front of a wall easier:
1. At the part, where you would usually jump over a piranha plant and stomp one of the 3 Goombas behind it, perform a jump standing next to the plant or kill the plant and jump over the 3 Goombas without stomping any of them. This will prevent two of the Koopas coming up from despawning.
2. Hit these 2 Koopas:

3. Jump behind the 2 Goombas and over the wall.

World 8-2

A perfect run will end with a high 342 on the timer, if you do not use the bullet bill glitch (343 is possible, but yields fireworks), and a high 337 or low 338, if you do use the bullet bill glitch.
If you are still Fire Mario, a save way to get a low 342 is to bump into the pipe, which forces a slowdown, and shoot the plant. To get the best framerule, you can slow down just so you don't hit the pipe, it's a bit more difficult and risky.

Several people pick up the mushroom in this level as it doesn't cost much time and gives you a free hit in 8-3.

A good bullet bill spawn can save up to 2 framerules (0.7 seconds) by performing a full flagpole glitch using it.

World 8-3

In this level you need to slow down to prevent the fireworks. You should finish with 242 left.

This level seems to be trivial as Fire Mario, but you can cause problems, if you are not careful. Don't wait too long with shooting the Hammer Brothers, they may jump directly on you as you pass them. Don't shoot them too early either: If they die that way they can throw a hammer to the right, which can be hard to dodge.

World 8-4

This is the only level where the framerule doesn't apply due to how the run is timed, so every frame matters.
Room 1
Just go right and enter the pipe. The piranha plant will mysteriously vanish before you touch it.
The big jump over the elevator has a 3 frame window. Instead you can jump onto the elevator, but it risks losing time, if you don't immediately jump off it again.
Room 2
This is the room with the walljump. When you run off the first pipe while holding B and right all the time, you only have 1 frame to make your next jump. If you briefly press left or just let go of right, you have more leeway at the expense of 1 or 2 frames lost.
The walljump is explained in detail on the main page and saves about 0.7 seconds, so seriously consider, if it is worth attempting. You will save about 5 frames if you do the walljump facing left instead of right.
Room 3
This room has a Wrong Warp glitch, but it is pretty easy to do.
Room 4
It doesn't matter, when Mario starts swimming as long as he doesn't reach maximum walking speed before you do it. If you hold right and touch the ground, Mario will be immediately reduced to maximum walking speed (2/3 of maximum swimming speed), so let go off right, if you are about to briefly touch the ground. Note that Mario will immediately lose all speed, if he hits the pipe without entering it and he will not enter it, if you don't hold right.
If you are still big, make sure to duck while swimming (see main page for details).
Room 5
As Fire Mario shoot the Hammer Brother, the small Mario method of just jumping on him doesn't work. If you get hit by the Podoboo, the invincibility lasts until the axe.
Theoretically it would be faster to kill Bowser than to take a hit from him (dodging him without a hit is of course the fastest), but shooting the 2 early fireballs instead of duck-jumping also places you at a higher risk of being hit by the Podoboo.

RTA WR and TAS comparison

TODO