Here is another web site for checking UK number plates: https://www.mycarcheck.com
Although this site does not tell you anything about the car/plate unless you pay them £3.50 for their "14 fields of vehicle information from the DVLA". It does give you the vehicle manufacturer. It can be used in conjunction with the DVLA web site. Sometimes you can't use the DVLA web site as you need Plate + Manufacturer. i.e. is a plate once registered to an MG now registered to another more modern car...

You might contact the person you bought the car from, and see if they might remember who the prior owner was. That might get you back one or two owners before people run out of memories.

You might contact your state DMV to ask about prior registrations for your vehicle. Many states do not keep records more than 10 years. Some states dispose of all records if the registration is not renewed for 7 years. Many states will not release contact information for a prior owner, so it may be slim pickings. It could help if you know someone who has access to police records and could run the check for you. If you do find this information, the next step back may be a registration from another state. Then you contact that state DMV and try again.

Putting your car's number plate number into an internet search engine can sometimes turn up hits on auctions sites.

If you have the chassis number you can contact British Motor Museum (Heritage Motor Center) to get a Heritage Certificate, which is sort of a birth certificate for your car. It will tell you the production date and original body and interior colors and any special order or optional equipment. It may also tell the original shipping destination, distribution center or the ordering dealer, and occasionally the name of the first owner if it was a special order car.

In the UK, the Kithead Trust holds many historical records of all sorts of motor vehicles, particularly a large quantity of motor vehicle taxation records.

The North American MGA Register has a database of several thousand MGA ranging from currently active to long since scrapped. No on-line link to the data base, so you have to contact them to ask.

The MGA Register of Holland has a database of a few thousand known current and perviously registered MGAs.
www.mgaregister.nl/register_prod_search.php

Australian Twin Cam Register:
http://users.tpg.com.au/bobles/Register/Register/main/reg.html

MG Car Club d'Italia: http://www.mgcarclubitalia.org
The next four links no longer work, as the club has apparently changed its domain name on the internet, and the new web site is not in the same structure. You can ask the club management. If anyone discovers where these web pages night be located in the new web site, please let me know so I can fix the broken links. Meanwhile, these links connect to archive files from the early 2000's.
www.mgcarclub.it/dati-mg/COLORI-MGA1500.htm - MGA 1500
www.mgcarclub.it/dati-mg/COLORI-MGA1600.htm - MGA 1600
www.mgcarclub.it/dati-mg/COLORI-MGA1500TC.htm - MGA 1500 Twin Cam
www.mgcarclub.it/dati-mg/COLORI-MGA1600TC.htm - MGA 1600 Twin Cam

Local car clubs nearly always keep records of their current members' names and addresses, and sometimes historical records of past members. The clubs also often keep records of the cars owned by the current members, and sometimes some historical records (not necessarily recording chassis numbers). While there are a LOT of independent local car clubs, it may help if you have some idea of the geographical locaton of your car at some point in the past.

Here is a list of every known MG and all-British car club around the world. It is a large list, and I try to keep it up to date.
www.mgaguru.com/links/mgclubs.htm

www.mgaguru.com/links/links_mgc.htm - MGA registers and clubs around the world.
www.mgaguru.com/links/links_mgp.htm - Dozens of personal MGA web sites.
www.mgaguru.com/links/links_mga.htm - MGA specific links.
www.mgaguru.com/links/links_mgx.htm - Other MG links with MGA interest.
This last one includes national and international MG clubs around the world. The national clubs can often lead you to affilliated local clubs.

Good hunting. This is a noble quest, but often leads to dead ends. If anyone has any other suggestions, do tell.

Addendum 03 Jan 08:
At 07:59 PM 1/3/2008 +0000, Roger Martin wrote:
- DVLA was set up in the mid-1970s, not in the early 1980s
- Only records of then currently licensed vehicles were trasferred from local licensing offices to DVLA for computerisation
- Records for cars already 'off the road' at that time were either transferred to local records offices or historical societies (e.g. Kithead Trust) or destroyed, at local licensing office discretion when they closed, in 1978 I think.
- A very useful book 'How to Trace the History of Your Car' by Philip Riden documented all the surviving UK vehicle record holders, but it was last printed in 1991 and is now out of print so only 'used' copies are available via the usual sources. Also, I guess some records may also have 'changed hands' since the book was published, or at least moved address.

FYI:
- Owners of unlicensed vehicles could 'reserve' their vehicles' registrations at DVLA up until 1980, after which they were (then) deemed unrecoverable.
- More recently, in conjuction with a vehicle clubs assurance process, DVLA will now reissue original registrations when a car is roadworthy and relicensed (not whilst still a wreck awaiting restoration).