The nightmare level of many any% runs. Bounce on the fish at the start of the level to skip the platform/flame melon. Next, perform an extended flutter by bouncing on a jumping fish in the water (make sure to hold down jump prior to bouncing, or you won't go high enough). This allows you to run along the top of the level and safely skip part of the level.
The next area is just tight jumps over slippery surfaces. Skipping super baby mario is the fastest route. The jumps through the ice blocks at 0:38 are especially scary; you may want to do a couple left-rights or flutter through this section to be safe.
In the next area, when bouncing on the snowmen, you must manually release jump and repress it to flutter: you cannot bounce on them and simply hold jump for an auto-flutter like most other enemies in the game.
Fast/easy snowman bounce: 
There are many ways to increase the speed of the skiiing section by jumping at the right times to avoid upward slopes or flying too long in the air, but they can be quite risky. There is probably an optimal and safe route waiting to be discovered. Check the Japanese runs on nicovideo, or the any% TAS on tasvideos.org for ideas. At 1:46 there is a frame perfect death if you jump right at the 3rd coin . Make sure you either jump a little before or after this coin so you don't die.
Room 2 comparison 
Noob Strats for spike clearing
The video on the right shows two strats that are easier & safer to do then the fast strat where you keep running and shot the spikes. And the best part is that both strats only need 1 egg. It is advised to just learn the first strat even though it is the slower one as it is really easy & safe. Once you are better & more comfy at the game just move to the fast strat. But feel free to try out the 2nd noob strat if you want. Good luck on not dying!
Alternate strat - penguin push: 
Alt. 2 - penguin bounce to perfect jump to extended flutter: 
Alt. 3 - fire melon item to clear an ice block:
Risky/backup strat - duck/slide; no eggs: 
Fast strat (semi-risky): 
Yolo strat: 
You can also just throw 2 eggs if you have them.
Visual Cue for first strat:
The idea here is to have the top of the cursor be aligned in a way where it sits right in the middle of the background line. A little bit higher/lower should work too... just mess around with it and see what works for you! If you don't get a good alignment right away just keep cancelling your egg throw and try again until you got one that you can be sure that it works :)
Visual Cue for second strat:
(Note: The second strat has a 10 frame window)