What does Detect Magic show for different Warlock special weapons?

Question

For reference, Detect Magic snippet:

you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.

Warlocks have a few unique weapon options. I am interested in what exactly does Detect Magic show for the following:

1. A weapon created via the basic Pact of the Blade feature
2. A magic weapon transformed to Pact of the Blade weapon (via the ritual), compared to before the ritual
3. Pact of the Blade's Improved Pact Weapon (eldritch invocation) and its created weapon with +1 to hit and damage
4. Hexblade's chosen weapon, by touching a normal weapon after a long rest
5. Hexblade's chosen weapon, by touching a magical weapon after a long rest, compared to before choosing it

Answer 1

Pact weapons would probably show up as magical, but as for the school, if there is one, that would be up to the DM; "hex" weapons wouldn't, unless the weapon is magical already

1. A weapon created via the basic Pact of the Blade feature

These warlock class features call out these weapons as magical. From Pact Boon - Pact of the Blade (PHB, p. 107):

This weapon counts as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.

Whether this means that it is actually magical for the purposes of detect magic, this isn't clear, but this gives a DM something to work with to make a ruling. I'd personally rule that the pact weapon is inherently magical, but that isn't explicitly stated in the text.

Further reading: Pact of the Blade and Antimagic Fields

2. Magic weapon transformed to Pact of the Blade weapon (via the ritual), compared to before the ritual

As for your second point, the weapon that you transform into your pact weapon is explicitly called out as being magical before you do so (PHB, p. 108):

You can transform one magic weapon into your pact weapon by performing a special ritual while you hold the weapon.

Since it is already a magical item, it would show up as magical to detect magic, but it would have done before it became a pact weapon as well. Whether being a pact weapon has any impact on the school of magic it registers as would be up to the DM (see the last section of my answer for more on this).

3. Pact of the Blade's Improved Pact Weapon (eldritch invocation) and its created weapon with +1 to hit and damage

As for the Improved Pact Weapon invocation, we can turn to the general information on eldritch invocations (PHB, p. 107):

In your study of occult lore, you have unearthed eldritch invocations, fragments of forbidden knowledge that imbue you with an abiding magical ability.

That seems more clean cut to me; if an invocation is a magical ability, and the Improved Pact Weapon invocation affects the weapon, then the weapon is magical thanks to that invocation (at the very least, magical enough to show up as magical to detect magic, even if the magic being detected is the invocation, not the weapon).

However, this related question shows that not everyone views such as thing as so clean cut: Is a Warlock's Improved Pact Weapon considered a magic weapon?

4. Hexblade's chosen weapon, by touching a normal weapon after a long rest

The Hex Warrior feature of the Hexblade warlock says (XGtE, p. 55-56) doesn't mention magic at any point, so presumably the "magic" here is that you are more capable of using it that you "should" be, thanks to your eldritch magic, but the weapon itself appears to be no different. I would conclude that the weapon is nonmagical and you would get nothing from detect magic.

5. Hexblade's chosen weapon, by touching a magical weapon after a long rest, compared to before choosing it

Similar to the above, except that because the weapon in this case is already magical, it would show up as magical to detect magic, but no different than it would have done before the Hexblade got their hands on it. Given that I've concluded above that a weapon being a hex weapon doesn't change anything regarding what detect magic would pick up on, then the magical weapon's magic would not be altered by Hex Warrior in any way.

Regarding the School of Magic

At no point in any of the descriptions of these warlock class features is a school of magic mentioned, so if a DM were to wish to treat any of the above as magical for the purposes of detect magic, they would need to come up with a school of magic themselves, or conclude that there isn't one. For weapons that were already magical (as per #2 and #5), they might want to use that weapon for guidance, if it mentioned a school of magic, but to the best of my magic, most (all?) magic weapons don't typically do that, so once again, it's probably just up to the DM's judgement.

Further reading: Do magical items have a school of magic associated with them?

Answer 2

Detect magic indicates magic objects and potentially the relevant school

That means it can detect magic weapons, modified or not.

Hex warrior weapon

The hex warrior feature does not say anything about magic, indicating that a normal weapon cannot be detected and the magic weapon can be detected as before.

Specifically, it says:

mystically channel your will through that weapon

Indicating that the effect comes from the warlock rather than a modification to the weapon.

Pact weapons

As for the pact weapons, the normal ones would probably not be visible, while the improved ones are clearly magical, as concluded in the accepted answer to this question: Is a Warlock's Improved Pact Weapon considered a magic weapon? The question has other answers with different readings. Obviously, the answer to that question that one considers changes the answer here. However, once you have decided whether or not to consider pact weapons magical, the answer to the present question is clear: If they are magical, they are detected like any other magical weapon, otherwise they are not detected.

Schools of magic

Concerning the schools of magic, these are defined for spells but not for the features in question. Since the spell description says "if any", "none" would be a valid information and the right one for these features.