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I'm an ex-pat British Citizen living in New Zealand. Many years ago (c1987), before the days of the Visa Waiver Program, I went to New York for a short business trip (about 10 days), and was granted an indefinite visa.

I've never had reason to go back, and obviously, in thirty years my passport has expired and been renewed several times. I'm not sure I could find the one with the visa in now.

I may need to visit the US for family reasons sometime in the next year. It'll only be a short visit - 2-3 weeks - and I'll be returning home to New Zealand, either direct, or via the UK.

Based on a sketchy understanding of US immigration rules, if I've had a visa before, I'll need one again. However, as a UK citizen I would otherwise qualify for visa-free entry.

Should I try and get my visa reissued for my current passport? Should I apply for a new one? Or, given that the VWP didn't exist when my visa was granted, can I now just turn up and enter under the VWP?

I should probably stress that there are no complexities here. My original visit went exactly as planned, and my proposed visit is just to visit family - a holiday.

A prior visa does not disqualify you from using the VWP.

Among the eligibility requirements for the VWP, you must not have been removed from the US, and you must not have overstayed any previous VWP admission. 8 USC 1187(a)(7):

(7) No previous violation

If the alien previously was admitted without a visa under this section, the alien must not have failed to comply with the conditions of any previous admission as such a nonimmigrant.

8 CFR 217.2, Eligibility, section (b)(2):

(2) Persons previously removed as deportable aliens. Aliens who have been deported or removed from the United States, after having been determined deportable, require the consent of the Attorney General to apply for admission to the United States pursuant to section 212(a)(9)(A)(iii) of the Act. Such persons may not be admitted to the United States under the provisions of this part notwithstanding the fact that the required consent of the Attorney General may have been secured. Such aliens must secure a visa in order to be admitted to the United States as nonimmigrants, unless otherwise exempt.

Don't forget to get ESTA authorization, make sure you get it from a .gov website, and enjoy your visit.

As an aside, even if you could find your old indefinite-validity visa, it would not help. From the State Department's page About Visas - The Basics:

What are indefinite validity visas (Burroughs visas) and are they still valid?

Indefinite validity visas (Burroughs Visas) are tourist/business visas manually stamped into a traveler’s passport which were valid for ten years. Effective April 1, 2004, all indefinite validity Burroughs visas became void. Therefore, if you have an indefinite validity visa you must apply for a new visa for travel to the U.S.

Now I'm certainly unable to reconcile "indefinite validity" with "which were valid for ten years," but the next sentence is unambiguous: that visa has been invalid since April 2004.

(Edit: the ten-year validity period was applied to these indefinite-validity visas retroactively by a change in the Code of Federal Regulations.)

I also would not read too much into the statement that "if you have an indefinite validity visa you must apply for a new visa for travel to the U.S." I am certain that the writer simply didn't consider the fact that the VWP would also allow such a person to travel to the US. But if you're uncertain about taking my word for this, you can get in touch with CBP yourself. If you do so, please come back and post an answer to your own question.

I was originally born in Hungary and have recently obtained Canadian citizenship. Hungary "recently" (2008) became visa free and I personally know a lot of people (including my brother and mother) who travelled with just an ESTA who had a visa previously -- which everyone needed before.