"One Fateful Day" Side Missions

Many of the opponents you face in the two Festivals will have unique cosmetic items available as challenge rewards. You can still fight them after completing the side missions, but they might be a bit harder to find.

Speak to Alice, and you'll be entered into the Knock Out Festival.
Be careful travelling to Pao Pao Cafe after, as it'll immediately put you into a string of cutscenes and fights.

Reach Style Rank 9 with Terry, then speak to him.

You must finish the previous mission to enable this one.
Return to Beat Square and speak to Joe Higashi.

You must finish the previous mission to enable this one.
Return to Beat Square (yet again) and speak to Shopkeeper Biz.
Give the dog a flight ticket, then travel to Pao Pao cafe.

Reach Style Rank 18 with Terry, then speak to him.

After beating the first Knock Out Festival mission, return to Beat Square and check the poster beside your Hideout. Switch to Night and speak to Alice to enter the second set of fights.
After finishing, travel to Pao Pao Cafe again, then Aokigahara.

After finishing both Knock Out Festival missions, return to Beat Square and speak to Alice.

Reach Style Rank 9 with Mai, then speak to her.

Reach Style Rank 18 with Mai, then speak to her.

Note that you can only access most of the destinations required for this mission by first completing "Traveling with Mai!" and "Shiranui vs. Bushinryu!". The fights are really fun though, so I'd still recommend trying it, even if you can't finish it fully.
Go down a branching dock in Aokigahara, on the right and towards the screen, and you'll find a side path. Follow this path up to a statue, and make an offering of a Jaw Breaker Candy. Inside you can accept this mission to fight the five Ninjas within, each requiring a Ninja stone before fighting you:

• For an immediate stone, leave the cave, and speak to the Ninja just outside.
  
• Head to Beat Square (day), and there'll be another Ninja standing across the road from the subway entrance.
  
• On the left side of Fete Foraine, there is a child who will sell you a stone for 10,000 Zenny.
  
• On the far left of the Ranger's Hut, the old woman will give you a stone if you answer her quiz. The answers are Armadillo and Jaguar.
  
• On the left side of Enma's Hollow, there is a man you can fight for the last stone.

After beating all five Ninjas and completing the mission, fight the man in the cave and he'll give you a music track. Fight each Ninja again to gain a Lyric card, and he'll give you a second track in return.

Enter Beat Square and speak to Anike. Travel to Kenya, then back to Beat Square, and talk to Narumi.

The elephant picture can only be taken in a certain spot, follow Houma's gaze, and walk a short distance away from him. The camera icon should appear once you're in the right place. The picture can also only be taken during the day.

Reach Style Rank 9 with Elena, then speak to her.

This seems to only become available after the previous mission.
Travel to Beat Square, run north towards the giant staircase, and you'll find Narumi.

Travel to the Metro City park at night. You'll know the mission is available as there'll be a dialogue line upon arriving. Speak to Lisa beside the statue.

You can get into the back area of the concrete... Structure... Thing... By dropping down to the platform with the chest on, then walking into the grassy area behind it.

Reach Style Rank 18 with Elena, then speak to her.

Speak to Sandal.

Travel to Proud Spire, speak to the locals.

Fast travel to Nayshall's Central Bazaar, speak to Ti Khao Heeza.
I found Dee-Jay's air slasher to work well in catching drones that are far away. Look for rooftops with people sat on them for an idea of where you can stand to get enough height.
(You can change your equipped Master actions in the phone's Settings section)

You must finish the previous mission to enable this one.
Use the Nayshall hideout to switch to night, then find Ti Khao Heeza in the same place again.

Reach Style Rank 8 with Sagat, then speak to him.

Reach Style Rank 18 with Sagat, then speak to him.

In Proud Spire there's a stone lying around that you can interact with. It will only give you a reward once.
Get down to at least 60 logs in the Mae Mai Mud Muay Hard Mode challenge, and you'll earn an EX color for Sagat.

Speak to Art.
More alerts will become available as you play through, and you'll need to switch between Day and Night repeatedly to get them all, even after an objective is marked as completed. All the fights are pretty easy, so you can grind some Style XP for a lesser used style off this.
There are 13 requests in total.

Will be automatically accepted after dealing with two disturbances.
If you want to get it done fastest: Grace is the correct answer. Moving ahead of her will cause a fight and restart the sequence. She'll turn around at the top of the ramp to the lower floor, and after speaking to Old Man Lloyd. After that, you can follow her to the door and the next sequence will play.

A red drone will be spawned outside the building afterward, which will allow you to immediately teleport to the basement lab.

Reach Style Rank 9(?) with C. Viper, then speak to her.

Reach Style Rank 18 with C. Viper, then speak to her.