I have only seen it used in dahl dishes, presumably because of it's claimed digestive properties.
Asafoetida is incredibly pungent; anything more than a pinch seems too much and unpleasant.
I think of it as a flavour-enhancer, a natural, Indian MSG rather than a spice in the normal sense.
It makes your mouth water quite intensely and seems to stimulate the savory tatse buds (umami).
This is lovely when done subtley and kept in the background.
Therefore, I suspect you could add small pinches to all kinds of savoury food.
cooking.stackexchange.com/users/8791/YumYum
It goes well in just about any kind of curry-based dish in the styles of the Indian subcontinent, from Sri Lankan mutton rolls to biryani to traditional curries.
I've never seen it in tandoor-based dishes, but I don't get out much.
cooking.stackexchange.com/users/4976/James McLeod
In the recipes I've seen (and made)
it goes mostly in dahls and other dishes that cook relatively slowly.
I think it usually goes in about the same time you would put fenugreek in: After the vegetables and some water are already in the pot, not directly into the oil like cumin.
cooking.stackexchange.com/users/611/Carmi
For a good flavor, use it in yellow dals.
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I used to have a recipe for a Pakistani curry gravy that was heavy on the hing (asfoetida).
This was from a restaurant that used it as their base for many of their curry dishes.
Googling has failed to turn up any hits that look right
but I do get more and better hits by googling on hing than asafoetida.
cooking.stackexchange.com/users/7129/Wudang
Asafoetida or Heeng as known in Northern India should be used in lentils or any dish that has beans as its main ingredients.
Its anti flatulent but has strong odour, hence use with beans :-).
We use it mainly in lentils or Dals.
I remember its being part of South Indian pickle recipes as well.
Its must have in kitchen
cooking.stackexchange.com/users/5272/Kumar
You can use it in any dish as an alternative to garlic (but in smaller quantities).
To be honest though, I haven't been able to find a dish which indeed tastes better with asafoetida than with garlic. :)
cooking.stackexchange.com/users/5767/Domchi
Asafoetida is used by some Indian religious groups as a substitution for onion.
Unless you have a religious aversion to onion I would discourage this practice.
It is exceptionally pungent and has an onion-like aroma.
Recipes will call for as little as 1/8 tsp to flavor an entire pot.
The only application that I have found that I like it is in lemon pickles.
In that case I don't want chunks of onion and I want the flavor subtly and evenly distributed through the jar.
cooking.stackexchange.com/users/2001/Sobachatina
Asafoetida is a digestive aid or specifically anti-flatulence.
So you'd predominantly use it in vegetable, lentil and bean dishes.
It adds little in terms of flavour and like turmeric is used mainly for its digestive qualities.
cooking.stackexchange.com/users/14539/spiceyokooko
A Rogon Josh curry recipe I have calls for the meat to be marinated in a mixture of yoghurt and a half teaspoon of asafoetida.
I have tried the recipe both with and without and there is definitely something different, but I can't quite put my finger on it.
cooking.stackexchange.com/users/8836/Kaz Dragon
I have heard about the medical benefits of  Asafoetida .   I used it once in  Brinjals  (which while cooking contained only  Ginger  powder,  Fennel  powder, salt and  red chilli  powder). The addition of  Asafoetida  resulted in the diluted flavor of the remaining spices in  Brinjal  curry.  In which kind of foods does it make sense to use  Asafoetida ?
In which kind of food items can Asafoetida be used?
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cooking.stackexchange.com/users/6168/Aquarius_Girl
Asafoetida is a digestive aid, or specifically anti-flatulent. It is mostly used in dahls and other dishes where lentils or beans are the main ingredient. It is generally used with beans and lentils as it aids digestion.
It has digestive properties and is generally used with beans and lentils. Use a small pinch in all kinds of savory dishes. It works like a flavor enhancer. It can be a good substitution for onions.