I'm a writer, and I've been wondering how to incorporate comedy into my story. I aim to have moments where my readers can burst out laughing and be left mildly entertained after reading my chapters. What type of things make you laugh really hard and why? Is it shock comedy, where something appears suddenly without warning, or built up comedy, where there is tons of foreshadowing to lead to one very funny climax, etc.

Dark humor, death, blood, gore. 

Make it all funny. I want to laugh when MC's wife and kids step into a waste grinder. 

That's what I'm doing. Aleron Kong is very good at in The Land: Chaos seeds, series. 

both works. but i like references.
the incredulous looks of the readers when they finish reading a chapter is the best. 
the thing about comedy is suspense. if your lead-up to it is bad then the joke sucks. a bad running is also good if it keeps the plot moving.
for example

SackOfSnails Wrote: Dark humor, death, blood, gore. 

Make it all funny. I want to laugh when MC's wife and kids step into a waste grinder.

i would do this part into the story. with the blood pouring all over his face 

As a case of bad dream
and he wakes up to a dog peeing on him, and as expected, his face.

A lot of people struggle with this I think, partly because of the brain rot induced by japanese 'walk in on naked woman get smacked' school of slackstick 'comedy'. In my opinion that is not comedy at all. Misunderstandings, a dense character, puns, very bad puns and funny situations. Something more witty than, oh I walked in on this girl changing her clothes so I deserve to get knocked out and that's somehow funny.

Stuff like someone bathing in the river and their pants disappeared so they have to make their way back through the village without pants. They dodge behind barrels and hide from townsfolk and just when they think they are being super stealthy and made it back safely, they get exposed in a fun way like infront of an old lady or a child, something shocking. Then maybe the old lady raises her eyebrows and make a remark about how the persons ballsack looks like their handbag. Maybe the old lady is hard of seeing and grabs the guys nutsack, rolling it between her fingers as she remarks upon the strange texture of her handbag and hasn't it shrunk in size and whats that long sausage attached to it? How strange.

A girl is sleeping and dreaming that she's kissing her crush, but wakes up to find that she is kissing her dog or a toy or a shovel or whatever you think is funny. Be creative, use shock and the unexpected. Set up expectations for one thing to happen, then have it all come crashing down in an unexpected way. Never be predictable with comedy or it dies, as is clear if you've ever watched a Japanese romcom anime or read a romcom manga or LN. You know exactly what's going to happen when the guy walks in on a girl in the first or second episode of an anime 'kyaa hentai *punch*'. It's way too predictable, so I think being unpredictable and contrarian should be a focus when trying to write funny scenes.