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Interestingly, Cambridge Dictionary does not acknowledge the singular form in British English, it defines it as being uncountable

fog

A2 [ U ] a weather condition in which very small drops of water come together to form a thick cloud close to the land or seaocean, making it difficult to see:

• Thick/Heavy/Dense fog has made driving conditions dangerous.

• Mist, fog, and snow are common in this area.
• It took several hours for the fog to lift.

But in American English, fog is countable and uncountable. Unfortunately, the only example provided is the uncountable sense.

noun [ C/U ] US
[ U ] Heavy fog made driving conditions dangerous.

Longman Dictionary goes into some detail. To summarize, it describes fog as being both COUNTABLE and UNCOUNTABLE (mass noun)

• It will be a cold night, and there may be fog patches
• A blanket of fog covered the fields.

And lists the following phrases

a blanket of fog (=a large area of fog)
A blanket of fog lay over the town.

a bank of fog (also a fog bank) (=a large mass of fog)
As we approached the coast, we ran into a dense bank of fog.

patches of fog (=fog that forms in some places but not in others)
Patches of fog are expected later today.

1. There is a fog. 2. There is some fog. 3. There is fog.

All of the above are grammatical, although "fog" is something which you do not normally quantify. We don't normally say “There are some fogs.” Instead, we would use another word in front which is countable. For example, patch.

There are some patches of fog.

If the fog is extensive and covers a wide area, the term blanket may be used.

The hills were covered by a blanket of fog

I believe that sentence 1 is a type of ellipses for

1. There is a [patch of] OR [blanket of] fog.

What do the dictionaries say?

Macmillan Dictionary says

fog

[COUNTABLE/UNCOUNTABLE] a thick cloud that forms close to the ground or to water and is difficult to see through.

Oxford Living Dictionaries say

fog¹

1. mass noun
   A thick cloud of tiny water droplets suspended in the atmosphere at or near the earth's surface which obscures or restricts visibility (to a greater extent than mist; strictly, reducing visibility to below 1 km)

• ‘Your Jeep fog lights can help you cut through thick fog or rain with ease and without temporarily blinding your eyes.’

1.1 in singular
An opaque mass of particles in the air.

• ‘The cloud clearly isn't steam in the strict sense, nor vapour (these are both invisible) but a fog of small ice crystals.’

1. There is a fog. 2. There is some fog. 3. There is fog.

All of the above are grammatical, although "fog" is something which you do not normally quantify. We don't normally say “There are some fogs.” Instead, we would use another word in front which is countable. For example, patch.

There are some patches of fog.

If the fog is extensive and covers a wide area, the term blanket may be used.

The hills were covered by a blanket of fog

I believe that sentence 1 is a type of ellipses for

1. There is a [patch of] OR [blanket of] fog.

What do the dictionaries say?

Macmillan Dictionary says

fog

1 [COUNTABLE/UNCOUNTABLE] a thick cloud that forms close to the ground or to water and is difficult to see through.

2 [SINGULAR] a cloud of smoke or some other gas
fog of: a fog of cigar smoke

Oxford Living Dictionaries say

fog¹

1. mass noun
   A thick cloud of tiny water droplets suspended in the atmosphere at or near the earth's surface which obscures or restricts visibility (to a greater extent than mist; strictly, reducing visibility to below 1 km)

• ‘Your Jeep fog lights can help you cut through thick fog or rain with ease and without temporarily blinding your eyes.’

1.1 in singular
An opaque mass of particles in the air.

• ‘The cloud clearly isn't steam in the strict sense, nor vapour (these are both invisible) but a fog of small ice crystals.’

1. There is a fog. 2. There is some fog. 3. There is fog.

All of the above are grammatical, although "fog" is something which you do not normally quantify. We don't normally say “There are some fogs.” Instead, we would use another word in front which is countable. For example, patch.

There are some patches of fog.

If the fog is extensive and covers a wide area, the term blanket may be used.

The hills were covered by a blanket of fog

I believe that sentence 1 is a type of ellipses for

1. There is a [patch of] OR [blanket of] fog.

1. There is a fog. 2. There is some fog. 3. There is fog.

All of the above are grammatical, although "fog" is something which you do not normally quantify. We don't normally say “There are some fogs.” Instead, we would use another word in front which is countable. For example, patch.

There are some patches of fog.

If the fog is extensive and covers a wide area, the term blanket may be used.

The hills were covered by a blanket of fog

I believe that sentence 1 is a type of ellipses for

1. There is a [patch of] OR [blanket of] fog.

What do the dictionaries say?

Macmillan Dictionary says

fog

[COUNTABLE/UNCOUNTABLE] a thick cloud that forms close to the ground or to water and is difficult to see through.

Oxford Living Dictionaries say

fog¹

1. mass noun
   A thick cloud of tiny water droplets suspended in the atmosphere at or near the earth's surface which obscures or restricts visibility (to a greater extent than mist; strictly, reducing visibility to below 1 km)

• ‘Your Jeep fog lights can help you cut through thick fog or rain with ease and without temporarily blinding your eyes.’

1.1 in singular
An opaque mass of particles in the air.

• ‘The cloud clearly isn't steam in the strict sense, nor vapour (these are both invisible) but a fog of small ice crystals.’