Let's tackle this a slightly different way, whilst I've got time to go into more detail.

What Yvonne has referred to is the ability to create custom fields. However, these custom fields have limited functionality and are only available in certain areas. For example, you can't do a sort or search on a custom field. But that doesn't mean you can't achieve that. If you generate a report which includes the custom field, you can export to a spreadsheet and then sort or search the data to your heart's content.

It also means that the custom field information will not be carried through the stock management system. i.e. If you capture serial numbers as the stock comes in, you can't draw (or insist on selection) of a specific item indicated by its serial number. The work around would be to create a separate item record for each item, but that is going to be messy i.m.o.

Ultimately, you need to identify your critical control points.
If you are trying to identify lost stock items, you are going to have to record serial numbers at both entry to your stock (capturing bills) and exiting your stock (generating invoices).

If you are trying to identify who bought which item (identified to serial number) and when, you need only capture the serial number in a custom field on the invoice.

The alternative, as I mentioned first, is to purchase a POS (point-of-sale) or a stock management bolt-on system that has the functionality of tracking serial numbers from entry to exit of the system. This is essentially a parallel process which reduces duplication of data capture - a big part of going with integrated software solutions in the first place.

These bolt-ons do exist. It has become a lucrative support industry that adds custom solutions to the fundamental accounting platform provided by Quickbooks. But as pointed out, being quite specialised and (unfortunately) we need to purchase EU based versions as the USA bolt-ons are not compatible, they are fairly expensive in a typical micro or very small business environment. Quite commonly the EU version of an application is near double the cost of the USA equivalent.

The problem seems to be mainly one of scale - there are simply way more customers in the USA and amortisation of the development costs is spread over a bigger customer base.

There are two particular bolt-ons I've been looking at quite closely - one to handle multiple stock centers (which has the functionality Duncan is looking for among other things), and the other is a rather sweet scheduling and CRM solution.

To cut a long story short, there is a fairly simple work around using custom fields, but it would be more a parallel system than an integrated solution.