What should I buy with mission funds?

Short answer: depends

Long answer:

You have several variables:
1) How many MF you gain per day (I think cap is 135 per day with platinum APEX, 2 challenges and all other strike missions including APEX ones)
2) How many credits you earn per day
3) What state your manifest is in

Having done the math for my personal case (~120MF, 100k+ credits, common and uncommon pre-1.09 manifests maxed), I found that spending the MF on common weapon mods is currently the best choice as the scopeless mods are actually useful.

Uncommon characters are easy to max out with advanced packs as they are first in queue (you get them as priority over variants)
Rare weapons are faster to pick up from arsenal packs (at 100k income you get 2 weapons per day vs 1 for spending all your MF)
UR characters are tempting but for 1 mil credits you get 11 rares + guaranteed UR. In reserve packs its fairly high chance of being a character. Now if your rare manifest is maxed as well, then UR cards become much more appealing. However they still can be obtained through the credits (Which are only capped by the amount of time you have to play and not MF which have hard daily cap).

Once again YMMV based on the state of the manifest and income.

EDIT: Forgot about equipment.  That is always a priority, so keep some MF on hand for Thursdays.

I am in the same position, except I can't reliably pug Gold, so mostly doing Silver. But I play with pistols/AR a lot, so I think those common mods might come in handy, while I am grinding the uncommon pool. So, dunno.

My view on using Mission Funds is:
1. Equipment. Keep replacing with better ones (if starting new) or until you get the right one for your primary character that you can farm Gold with, then go with...
2. UR Character. As most UR characters can function on rank I. or until I hit XX with my primary character. Then I go with...
3. UR Weapon. Until maxed. Then finally...
4. Weapon Mods.

The other items can be obtained via opening boxes, so I feel that I'm wasting MF for something I can obtain easily already. As for the rare mods that do not drop from boxes, I do not feel this is an issue, as if you are that new to the game, you'll benefit more from opening boxes for the standard mods that can work just as well. And if you are late into your manifest, then you've still got UR's to complete. If the UR's are no longer a factor for you as your manifest is pretty much complete, then by all means the Weapon Mods. But until then, normal Weapon Mods should be more than suitable for farming runs.

ALL OF THE THINGS.

Probably neo-weapon-mods and UR characters.

I refuse to buy the mods with mission funds. Equipment, xp boosters, consumables, weapons and characters are the things I buy.

I too would like to know this.  I've seen one person say they saw someone get one out of a pack, but I haven't seen any proof.

But they say new mods can be purchased only with MF.

....I hope this changes in near future...

Totally legit way to increase longevity of the game.

I think they rotate. So next month instead of UR characters, it will be UR weapons.

I'm going to start saving up my MF. I'll buy new equipment on Thursdays, but that's it. Stupid me spent too much on First Aid packs in the past, before they added all this new stuff to the item store. I might buy the odd common mod here or there, but I won't touch uncommon mods for a while. After maxing out my uncommon characters and weapons, I went straight to 100k packs, so I still haven't unlocked all my rank X uncommon mods yet. They do pop in 100k packs once in a while though, so I'll get there eventually...

There are other weapons.