I've successfully made the Worldpay extension work in Abantecart 1.2.9.  I had to make a few changes to the code, and compose some new templates (and some dummy files) but it is now working.

I'd like to tell you how I resolved this issue. I in fact cleared down the website and re-installed AbanteCart. When I did so I noticed a message during the installation process which said make sure the PHP setting opcache was turned off. So I made sure that was the case.  This also required the setting zend_guard_loader to be turned off. On doing a clean install, I still got the error messages.  Therefore it was clearly a hosting issue, not Abantecart, so I thought.  By playing around with the PHP version on the Hosting Control Panel the problem went away. 

However recently the problem returned when going into the resource manager. This was after loading a large number of images. The hosting Control Panel error messages were complaining of using too much virtual memory. This was happening when going into the resource manager. Presumably at this point all the images were being cached. So I turned off caching for images: 

System -> Cache - check images and click on 'Clear Selected Cache'.

The annoying error messages have gone away, I hope for good.

Abalabo:
The memory_limit I have set to 384M in my little php.ini file and that is what I see in the full php_info report that you referenced (System -> Settings -> System and then PHP info button top left). From what I've seen I think this is way over the top.

Barasa:
I have contacted my hosting provider and I have given above the information that they provided. They seemed unable to help further.  It almost looks like a bug in PHP, but this has only recently started happening.  What I said was:

Looking at the last 30 days, there have been periods when (14-15, 22-25 Nov) when the virtual memory limit has been reached, but I have only experienced these error messages 'Internal Server Error' in the last day or so.

This makes me think the answer lies in the code of Abantecart which is way beyond my understanding at the moment.

Another thing I tried was to 'switch off' the Foxy template which I bought and installed some time ago and I think recently has been automatically updated. This made no difference. 

Could this possibly be the problem?
The only substantial change I made in the short time before this started happening was to change the code of the 'Royal Mail' shipping extension (extensions/default_royal_mail/storefront/model/extension/default_royal_mail.php)  and to add a new shipping extension 'Yodel' which was modelled on the 'Royal Mail' one.  Now I added that extension simply by adding the new folder 'default_yodel' with all the necessary files (which I had previously debugged in my offline copy of AbanteCart on my local machine) to the extensions folder.  I had logged out from the control panel before doing this.  My understanding was when the extensions page was brought up, Abantecart would find the new extension and add it to its internal data (this extension makes no SQL changes) and everything would be fine, and it was. The extension worked (apart from some language 'translations') and the desired extra shipping options appeared in the checkout. Soon afterwards I started getting these Internal Server Errors.

So one of these could be the suspicious culprit, right?

Ok, so I uninstalled both these extensions and deleted them using the buttons on the extensions page. I then physically moved the default_royal_mail and default_yodel folders to be under the tmp folder, out of the way. AbanteCart shouldn't see them there.  So these extensions cannot in any way be responsible for these errors. But I still get them.  Is it possible that some residual entries in the database might remain which could be throwing a spanner in the works?  (Well I found some in the dataset tables, but these seem to be recording the installing and uninstalling of the extensions. It's hard to think these could be responsible.)

Is it possible that the database has become corrupted? 

Do I have to do a complete reinstall?

Any other suggestions?   Would my sending you a verbose debug log help?

Thanks for your attention so far.

Well, I've been able to solve this myself. I hope the below helps others.

I've discovered that the extension was invoked, but failed to show anything on the shipment drop-down on the cart page for two reasons:

- no location zones had been set up for the location (country) which was selected in the Shipping Extension admin page
- the cost was calculated (correctly or incorrectly) to be zero.

The xx_zones_to_locations table only had entries for the USA, whereas UK entries were essential given that this is a UK-only service, the Royal Mail.  I don't know if I had installed Abantecart correctly, but by tinkering directly with the database, I have added 100 entries for UK locations to this table. 

I think you can achieve the same result without knowing any SQL by selecting System -> Localization -> Locations. Make sure the country is there, click on the Edit symbol and choose Location Zones. If no zones for this country are shown, click on the plus top left.  Select the required country, select all and save. The zones will be added to the 'location'.

I have to say that this ability to choose 'zones' within countries is a very good facility of Abantecart. Without it, getting the charges right for shipping to remote locations would not be possible.

However, this was not enough to bring up the shipping in the shipment drop-down.

In addition to this, references in the code to weight units were incorrect, the string literals should end in ..._class not ..._class_id.  This meant that weights in grams were not converted to kilograms, so the code tried to find a charge for a weight of hundreds of kilograms! No surprisingly this failed, the maximum weight being 1 kilogram, and so a cost of zero was 'calculated'.  When the weight classes were corrected, the shipping offer appeared in the shipment drop-down.  Success.

I attach the corrected php file.