Chrom vs Falco

This MU can go both ways very quickly. If chrom SH nairs, watch his landing position. It's very easy for him to space it so it's safe. If he lands behind you, mix up between jumping immediately and holding shield; he's going to ftilt, fsmash, or jab to poke your shield and catch your timing. If he lands far in front of you (tippers it), there's not too many options. Just get away or try and fair OoS or such. If he's right in front of you, you can shieldgrab him if you do it immediately.

This is the worst part of the MU. If Chrom plays smart, he'll be ftilting and catch any quick options with fsmash. Do neutral or jump getup (roll is abysmal against Chrom), be very smart about side B (if he reads it, you will die), and jump to a platform if possible. Get back to center stage slowly but safely. If he catches you, you'll start over in this same situation again.

Chrom will destroy you if you land near him. Play patiently, land on platforms and watch/shield, and wait for him to let you land somewhere where he can't cover.

Chrom will ledge trap falco incredibly effective, but falco can't ledge trap in return. Play smart, hold center stage to do damage and eventually edgeguard Chrom.

Chrom's initial up B has super armor on it. The best way to ledge guard him is actually to not go offstage much. Catch his double jump with laser (ideal) or an aggressive fair, and if he has to up B, then space a dair for the kill. It takes practice but ideally you should be able to hit Chrom's up B (at the top) with a sweetspot dair every time. If he uses up B and doesn't sweetspot ledge, and you're too far away to intercept it with dair, you can hit him out of it with a well placed laser to force him to up B again, and catch him the next time.

Chrom gets juggled hard. This is how you win the MU- play smart but apply pressure, bait airdodges, and keep him in the uair vortex as long as possible. If he slips past you, that's very bad- he can counter juggle nicely, so just play safe and don't let him land before you.

Watch out for Chrom's ftilt, fsmash, nair, fair in neutral. They're all generally quite safe and have some kill power. Dash dance to avoid them, go for grabs when you can (Chrom won't be shielding much but it's still the best combo starter you have). Poke with ftilt occasionally. Dtilt if you're behind him and he's shielding. Utilt or usmash as an anti air, but that's more niche. Overall, this MU is won by juggling more and landing edgeguards consistently.

Chrom's main win condition against Falco is keeping him in disadvantage. Choose a small map against procedural and safe players to force close interaction and more combo potential. Large maps for aggressive players so you can dash dance more/bait options.
Overall, Falco benefits primarily from platforms. Battlefield is his best stage (in this MU and overall), and FD is his worst.