Sorry if this is too obvious to mention, but you have tried googling "product name serial number" haven't you?

@Dave I assume b serial number the OP means the one which is unique to each individual item, not the bar code which applies to the product generally.

As do I. I tried for the first one I found, and saw this page. It would be worth trying given the amount of time already spent.

@Dave These are Apple Ear Buds. The serial number is on the item itself and the charging case, both of which were lost, hence the insurance claim.

You may be able to find your serial number in the Bluetooth information on a device you’ve paired with your AirPods: support.apple.com/en-us/HT209371

What indications do you have that the retailer ever had matched the serial number to your transaction? Are you sure the item serial number is located on the outside of the retail packaging? Did you see the retailer record the serial number during your transaction? Even if the retailer has records of all the serial numbers which they purchased, there's nothing here which indicates that they even recorded what serial number was sold to you. A retailer could, reasonably, operate on merely recording all serial numbers which they purchased, rather than which a customer of theirs purchased.

I'd also note that unless there's a regulatory requirement, there's not a business requirement for them to know what serial numbers they purchased, let alone match it to a customer. Not knowing that does leave them open to a customer returning a different identical item which was purchased elsewhere, but for which they also purchased one at their store. While that does happen, it's a relatively low occurrence issue, which the retailer may have chosen to just accept as a lesser cost than dealing with tracking such serial numbers for every customer, or even just for their own purchase.

@Makyen when I worked in audit many moons ago, it was standard practice to trace an item, by it's serial number, from the point of purchase to the point of delivery, for companies such as this. I never worked on the Argos account but it was standard on Dixons (owner of PC World Currys) and if such a trace was not possible it would have been a massive red flag. Maybe things have changed since then, but I would have thought that with today's technology that would be even easier now. And as for the serial number yes I am sure it's on the outside of the packaging (it says so on Apple's website).

@RobertLong There are advantages in a reasonable number of situations to different levels of diligence in tracking serial numbers. There are also costs associated with those levels of diligence, both in necessary infrastructure (upfront cost) and in employee time (ongoing cost). What level of diligence to use may be something which a company chooses on a product by product basis, probably chosen based on the relative cost of the item vs. the cost of more diligent tracking. From what you've said here, there's really nothing that indicates the retailer ever associated the S/N with you.

@Makyen, someone here is complaining that when they took back a faulty item Argos told them the serial number didn't match. There are other stories like that in the internet. I have just called them again, and they told me that if I had checked within 3 months they would have the details including the serial number, but because it is after 3 months, not only do they no longer have a record of the serial number - THEY NOT LONGER EVEN HAVE A RECORD OF THE ORDER. / cont....

.....and when you log into your "online account" any orders older than 3 months have disapearerd - there is an option to seach for "recent orders", which they told me on the phone means within 3 months, and that's it.

As an example, Apple seems to be switching to random serial numbers, so any fraudsters cannot just make up a number and hope it's a valid serial numbers. So they would at least know the exact serial numbers of any product that was ever built.

I know of very few cases where a retailer records the serial number of a product. At best they only record the product number or equivalent. Unless it is a very expensive item (like a car) they just have no reason to go to the expense of recording such information.

@RBarryYoung then why do they say they can only do it within 3 months of the order ?

@RobertLong Did they say that they had the serial number at POS (Point Of Sale) or did they just say that they don't keep order details longer than three months? Things may be different in the UK, but in the U.S. retail sales order details rarely ever include the serial no. It's not in the Bar Code and they sure aren't paying anyone to stop in the middle of checkout to find and write down all of those numbers. There's just no reason to do it.

@alephzero Sadly this is likely true, after-market insurance is almost always a bad deal. For one thing real insurance (like home-owner's or renter's insurance) will often cover such losses anyway (possibly with a deductible). Retailer's aggressively push after-market insurance on their sales because it is almost 100% profit for them, they pay nothing upfront, get an almost cost-free cut of the payment and then because the "insurer" is an unrelated company, they have no costs after the sale. It's a big win-win for them.

@RobertLong You will also note that at the example you give, Argos never checks their sales order-detail for the printer's serial number. Rather, they only check that the returned printer's serial number matches the serial number on the box that it was returned in. So there's still no evidence that they ever recorded it, and in my 40 years as a developer of many commercial, retail and inventory software systems, I can tell you that the serial number was almost never recorded anywhere.

@alephzero yes I do think the supermarkets record the sell by dates. They use this info to prepare a manifest of items that need to have price reductions when the items get near to the sell by date. They just print it out and go round the store sticking on the new labels. They don't have to go looking. This is the case at Ssinsburys, as I know the local store manager.

@RBarryYoung the serial number of the item is on every invoice for every product I have purchased from Apple directly. This is the first time I've bought an Apple product from a third party.

When the UK left the EU, did you get rid of the European laws on consumer protection? (a 2-year legal warranty where you deal exclusively with the vendor and nobody else)