Both I think. Neither sounds strange anyway...

Stick

"It's starting to snow."
"Is it sticking?"

"It snowed last night, there's about an inch on the ground."

"...While the snow lay round about deep and crisp and even..."

I'm with jinx. Both.

Small amounts "stick".

Once it gets over an inch-and-a-half, it lays.

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Neither. It settles.

When it just comes down and accumulates, it definitely "lays" ... if it "stuck", it wouldn't be able to blow into drifts. Sometimes, though, it adheres to a wall, or other vertical surface, so in that case, it would "stick". But I still always say "lay". I'm not really concerned with the snow that "sticks" to a vertical surface, I'm concerned about the snow that "lays", because that's the snow that I have to fucking shovel!

We have an old joke with my 94 year old grandmother...when it snowed she would call my brother and tell him "the ground's covered" whether it was or not. Her theory was that it's always covered with something...grass, dirt, asphalt. So my brother keeps up the tradition and calls her every season when we get significant snow.

We say the snow "sticks" around here, meaning it isn't hitting the ground and melting away.

The snow lies. Lays is ungrammatical.

Unless maybe that's how you get your eggs in the winter.

In which case being the omelette fiend I am, I might want to move there.

Maybe

Maybe not

From your second example: Snow lay thickly over the fields.

I've never even heard "lay" for snow.

Here in beautiful La Crosse on the Mississippi, we say accumulate, and we know a lot about about SNOW

don't know much 'bout trigonometry, but I do know...
 

The two are confused because of the conjugation of the two verbs:

lie: lie, lying, lay, (have) lain
lay: lay, laying, laid, (have) laid

Lie: to be in or move into an esp. horizontal position on a surface:

The mechanic was lying on his back underneath my car.
The cat just loves to lie in front of the fire.
She lay back in the dentist's chair and tried to relax.
Lie still a moment, John.
He lies awake at night, worrying.
He lay down on the bed and cried.
I usually lie down (=rest/sleep) for an hour after lunch.
Snow lay thickly over the fields.


The verb lay means to put (something) in esp. a flat or horizontal position, usually carefully or for a particular purpose:

She laid the baby (down) in its cot.
I'll lay your coats on the bed upstairs.
Perhaps we should lay paper over the floor while we're decorating the room .
She laid aside her book and went to answer the phone.
The dog laid its ears back (=put them flat against its head) and howled.
He laid down his knife and fork, saying he couldn't possibly eat any more.
Lay the rug flat on the ground.
There will be some disruption for the next few weeks while contractors lay a new cable/sewer.
We're having a new carpet laid in the hall next week.
They've been laying bricks (=making a wall with bricks) for two weeks and the first floor is already finished.

As we are not "putting" the snow into a horizontal position, the verb "lay" is correct, not as the original unconjugated "lay" but as the conjugated tense of "lie" as shown above.

So lay is correct, though in a sense UG was correct in that it is a form of the verb lie.

Clear as mud?