Informative article, more on the problem than solutions:

https://www.nextgov.com/ideas/2021/05/racing-sun-protect-america/174029/

A modern day Carrington event scares the heck out of me. Enough so that I have a metal trashcan Faraday cage filled with assorted radios, old computers, etc as backup in case it happens.
-Blast

A modern day Carrington event is indeed a cause for considerable concern.
It seems unlikely to have any effect on portable electronics though. A solar storm induces damaging electric currents in LONG electrical conductors such as power transmission lines, telephone lines, pipelines, railroad tracks, and even long wire fences.

LONG metal conductors can be damaged as can articles connected thereto.
A railroad track should survive, but signals, and switch motors attached thereto might be destroyed.
A LONG metallic pipeline should survive, but fires may be started and controls and instruments destroyed.

Note that only LONG conductors are affected, miles long at least. Conductors inches long inside anything portable should be unaffected.

The consequences can be reduced by insulating inserts or joins in pipelines and railroad tracks.
When erecting a long wire fence, use the wire in the lengths in which it is supplied WITHOUT JOINTS. At the end of a roll of wire, fix it to the post, and fix the start of the new roll to the post some inches away. DO NOT twist or crimp the two together.

Storing portable electronics in a faraday cage is prudent in case of an EMP which a different phenomena.
An EMP is a result of a high altitude nuclear explosion, and induces damaging currents even in very short conductors, such as those within portable electronics.
A metal trash can gives SOME protection but only partial protection due to the loose fitting lid. I would prefer two layers of protection, for example placing items in a metal tin, and then placing this tin in a metal trash can or a metal cupboard.

The last time I tried that, my wife hauled it off to the recycling place.

I have a role of aluminized tape to seal the lid to the can if things look hot between nuclear nations...but getting it wrapped in time will take some luck.
-Blast

Should the trash can Faraday cage be grounded? Or not? Thoughts?

Grounding would not be required. I would make sure that lid and can are connected e. g. with a copper band like cars use for connecting ground for the battery.

I dont believe that it matters if a Faraday cage is grounded or not. The basic principle is that within a Faraday cage, there can be voltage difference between any part and any other part, and therefore no electrical damage can result.
The contents do not "know" what if any voltage exists between the cage and the ground.

I suggest copper tape with conductive adhesive like I used to shield my cheap electric guitar from internal hum.

Conductive adhesive has to be checked for as that's not necessarily what's on ducting tape.

Would an old microwave oven serve as a Faraday cage?

An old microwave might work okay.

Quick test for the non-radio-inclined: Connect to your phone with a Bluetooth headset or similar device. Unplug the old microwave (so that no matter what happens, it doesn't get turned on during the next part). Put the phone in the microwave and close the door. If your Bluetooth device still works, it's not a great Faraday cage.

I would favour TWO layers of metallic protection, this need not be expensive or elaborate.
I use metal tins that previously contained biscuits, and these tins placed inside a metal filing cabinet.

Items that I store include

Two way radios.
Radio receivers.
Geiger counters.
LED light bulbs, various types including flashlight bulbs and vehicle bulbs.
LEDs, bare components not LED lamps.
Silicon rectifier diodes.
Electrical test meters.

Cellphones and PCs are in my view a lower priority as cellphone service is unlikely to be available, nor is an internet connection or enough power to work a PC likely to be available.

Zinc carbon batteries should survive just fine as should alkaline types and lead acid batteries.
Any more modern type of battery that incorporates electronics will probably be killed by an EMP.

It is worth storing the materials to make basic batteries, the raw materials keep forever and are invulnerable to an EMP.

The most critical components of a Faraday cage are continuous conductive material around whatever you're putting inside it and an insulation layer (a layer of cardboard will do, plastic or air works well too). There's a school of thought that suggests another layer of continuous conductive material on the outside, insulated from the inner layer. Any devices being protected should not touch any of the conductive material.

A large steel toolbox (often called a "job box") could work as the outer layer, although I'd be careful to make sure that the lid was tightly fitted and well-connected electrically to the main body.

My expectation is that something like a Carrington Event would damage or destroy large electrical grid transformers, along with anything else connected to long conductive wires. Smaller grid transformers connected to shorter wires may or may not be affected. Based on what I've read -- I'm certainly not an expert -- complex electronics within cars and small devices are more likely to survive than not.

Certainly, a close, strong EMP would damage or destroy almost any electronic device.

I very largely agree, except that I see no need for an inner insulating layer. All parts of a Faraday cage will be at the same voltage and it matters not if the contents touch the inner walls.

A steel "job box" would give a good degree of protection but I would still prefer a second inner container as a second line of defense in case the hinged lid of the outer box gives imperfect protection.

These job boxes are often used to protect expensive tools against theft. Much better than a trash can in this respect. Security is good, no point in protecting supplies against EMP if a thief takes them instead.

A utility transformer connected to long power lines would probably be destroyed by a Carrington event and would almost certainly be destroyed by an EMP.
A spare utility transformer in the power companies stores and not connected to anything should survive a Carrington event.
A spare utility transformer in storage MIGHT survive an EMP if prepared by connecting all input and output connections to each other and to the steel tank. The tank IS a reasonable Faraday cage and connecting each end of each winding to the tank SHOULD prevent damage. Utility transformers are designed to withstand much greater voltages than domestic electronics.

The inside of my metal trash can is lined with a blue, closed-cell sleeping pad so nothing touches the metal. It still needs a better air-gap seal, though.
-Blast

I don't think that's correct. I think that when an electronic device touches the conductive layer, that layer stops being a Faraday cage and starts being an antenna.

Would one of those 5- or 6-mil 'contractor' trash bag act as an insulator?

I expect it would work.