To many people, winter-sun cruises mean one thing – the Caribbean. As the Med cools down, dozens of ships descend on the West Indies: in some parts of the islands, you can’t move for mega-liners and the mega-crowds they disgorge into every port they visit.

Now, the cruise lines have discovered an alternative happy hunting ground in our winter: Australasia. Auckland, for instance, used to count itself lucky to get a dozen visits from cruise ships in a year. Last winter, it had 75.

I flew to Sydney and boarded Holland America’s Statendam, which was spending its third season down under. At a little more than 55,800 tons, it’s a modest-sized ship by today’s standards, and manageable, too, with most of the public rooms on just two decks.

A series of small lounges means you are seldom aware of the number of other people around. Only occasionally, in the self-service restaurant and when you see the collection of coaches needed to run the excursions ashore, do you remember that you’re one among a population of 1,250.

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To escape any hint of a crowd, I could always retreat to my generously sized “ocean-view stateroom”. Stateroom, note, not cabin: there is more America than Holland about Holland America, although it sails under the Dutch flag and has a preponderance of Dutch-speaking officers.

The onboard currency is the dollar, the culture American. The more intimate Pinnacle Grill has great haunches of prime US beef on its dinner menu for a £17ahead supplement. There’s no danger of anyone leaving the ship hungry.

“Thank you for coming,” said Geraldine Doyle, the Irish-Australian comedienne, in her stage act. “More important, thank you for staying, because God knows there could be food somewhere.”

Besides Geraldine, we had a magician, singers, dancers and Canada’s champion juggler in the Van Gogh show lounge. That’s Van Gogh to rhyme with banjo, not Van Gogh pronounced like standoff. There were four formal evenings. American passengers – the majority – put on tuxedos or, at least, jackets.

The Aussies, who have a looser concept of formality, swapped shorts for chinos. “It’s a constant debate in the company, although passenger surveys still show that people like the chance to dress up,” said Martin Groenendijk, the culinary operations manager. The fact that takings in the casino and bars tend to go up on formal evenings might also have something to do with it.

WE SAILED at 7pm, parading down one of the great waterways of the world, under the arch of the Harbour Bridge, past the open beaks of the Opera House, through the flurries of yachts and the bustle of ferries, then out through the Heads into the Tasman Sea. We were bound for Melbourne, but it took a day to get there, a day of onboard activities – a day, if you wanted, on the companion way of self-improvement.

As in t’ai chi or Scientology, there seemed an infinite number of levels to attain, starting at the mouth. On Statendam, travel not only broadens the mind, it whitens the teeth. “A cleaner, whiter smile can be yours today,” promised the ship’s daily newspaper. “Gentle Touch Teeth Whitening, $199, provides a brighter more vibrant smile.”

You could learn how to mix martinis (£6), “set a perfect table” and attend a wine tasting (£9). Other days offered dance classes, cooking lessons, more wine appreciation and advice on starting an art collection, with regular picture sales to get you going. It was pure Martha Stewart.

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