So the Halfling have a perk called "Lucky" that is as described:

When you roll a 1 on the d20 for an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll. 

So it doesn't say that it can only happen about 3 times a day or anything, so can Halflings effective reroll any 1 never critical fail unless they get two 1's in a row?

Yup. They're just that lucky.

Technically there are no critical fails in 5e, but I always tend to houserule them in.

It could save your bacon when it comes to death saves.  Remember a rolled "1" on a death save equals two failures but a rolled "20" on a death save puts you back at one hit point and stable.

Would this apply to spell attack rolls?

Lucky works with any attack roll. An attack roll made as part of a spell attack is still an attack roll.

You can’t reroll on initiative though...

Quote from john_frog >>

You can’t Errol on initiative though...

 Yes you can, initiative is an ability check and thus can be rerolled on a 1.

If you're a halfling / druid using a shapeshift and take 1 on D20, you're under the same rule Or not?

I googled it real quick, and I saw this : 

"You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so."

So yeah, you'd still have the option since it's not something tied to your character's physical body so much as like... their essence.

Quote from Farwulf >>

So the Halfling have a perk called "Lucky" that is as described:

When you roll a 1 on the d20 for an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll. 

So it doesn't say that it can only happen about 3 times a day or anything, so can Halflings effective reroll any 1 never critical fail unless they get two 1's in a row?

 

 Yes, they can re-roll every attack/save/check, the only limit is they have to use the result of the re-roll.

Here's a question: many staves or wands with charges say that if you expend the last charge you roll a d20 and on a 1, the item is destroyed. Would this also be affected by halfling luck? If so, a halfling could always safely drain the item without worrying about risking its destruction.

I would assume not since I don't think this counts as the player's check or saving throw. 

Quote from MirkinGraybush >>

Here's a question: many staves or wands with charges say that if you expend the last charge you roll a d20 and on a 1, the item is destroyed. Would this also be affected by halfling luck? If so, a halfling could always safely drain the item without worrying about risking its destruction.

I would assume not since I don't think this counts as the player's check or saving throw. 

 The trait is restricted to Attack rolls, Ability Checks, and Saving Throws.  Seeing if a depleted Wand gets destroyed or not isn't one of those things.

—Edit: the following is incorrect. Skill checks don’t exist in 5e, only ability checks.—

Here’s something that everyone forgets about: ability checks are different from skill checks. And halflings don’t get the re-roll on skill checks.

Rolled a 1 on that stealth check? Sorry halfling rogue, that 1 is gunna stick this time.

Quote from GlumPlum >>

Here’s something that everyone forgets about: ability checks are different from skill checks. And halflings don’t get the re-roll on skill checks.

Rolled a 1 on that stealth check? Sorry halfling rogue, that 1 is gunna stick this time.

In 5e there's no such thing as a skill check. A Stealth check is a Dexterity ability check and Halfling Luck absolutely works on it.

There's a lot of text about this in Chapter 7 - Ability Checks - Skills but this is probably the most appropriate quote:

... proficiency in a skill means an individual can add his or her proficiency bonus to ability checks that involve that skill. Without proficiency in the skill, the individual makes a normal ability check.

Well shoo, I stand corrected. Thanks for quoting that. I guess I was operating based on definitions from previous editions of the game. Skill checks are indeed non-existent in 5e.

Halfings with the lucky feat are terrifying. Halfling bards with a few levels of divination wizard and wild mage are even worse.
http://purplelizardman.com/2018/01/27/paragon-of-luck/ 

I know that halflings don't normally get to use great weapons, but if a halfling takes the great weapon fighting style, does their lucky trait add to the reroll ability for it? Like, turn it from reroll 1s and 2s to rerol 1s 2s and 3s? I am making Lucky into a feat for my next homebrew campaign and I want to know about all the little details.

Quote from githbrains >>

I know that halflings don't normally get to use great weapons, but if a halfling takes the great weapon fighting style, does their lucky trait add to the reroll ability for it? Like, turn it from reroll 1s and 2s to rerol 1s 2s and 3s? I am making Lucky into a feat for my next homebrew campaign and I want to know about all the little details.

 No, because the Great Weapon Fighting feat allows a player to reroll 1's on the Damage Die, but Halfling Luck can only apply to the Attack Roll. So if you're somehow playing a Halfling with the Great Weapon Fighting Feat and you roll a 1 on your d20 in your attempt to hit an opponent with a Greatsword, you can choose to reroll the d20 using Halfling Luck. If the new roll would hit the opponent and you're attacking with 2d6 damage, if either of those D6's come up as a 1, you can choose to reroll that d6 with your Great Weapon Fighting.