Similar questions have been asked on this subreddit before. However it should be noted that Shackleton’s ship, the Endurance, actually had a 350 horsepower steam engine and was not entirely powered by wind. It also boasts a massive wooden bow 1.3m thick and many ribs designed to strengthen the hull and provide resistance to ice it may encounter. The ship didn’t actually sink from hitting ice but becoming trapped and eventually crushed after it was trapped in an ice field.
https://www.coolantarctica.com/Antarctica%20fact%20file/History/antarctic_ships/endurance.php
When images were published earlier this week showing Endurance at the bottom of the Weddell Sea, it was hard not to think of another famous casualty of the ice that launched the very same year. When Titanic sank in the North Atlantic in April 1912, Endurance was still under construction in a shipyard near Oslo; yet the design of Endurance - wood-built with three masts and rigging - seems to place her in a far more remote age of seafaring. These were totally anachronistic features for pretty much every other maritime context, but polar voyages involved specific demands that justified them to the threshold of WWI and for a few years beyond. Even so, Endurance was part of the final generation that used them.
Why wooden hulls?
The conditions were exceptionally challenging: changeable weather, apocalyptic storms, the absence of passing shipping or resupply options, and most of all the ice. Even in high summer, polar seas could be a vast mosaic of floating ice. As the weather cooled, it had the potential to solidify into vast, slow-moving expanses of pack ice stretching hundreds of miles. Whalers (the most common vessels traversing the Arctic and Southern Oceans) usually spent polar winters in a sheltered cove where they knew the sea was unlikely to freeze, and sailed out again when the surrounding waters thawed. However, given the vast distances crossed and the difficulty of predicting conditions months or years in advance, scientific expeditions were always at risk of getting stuck in unfamiliar seas if the winds and waters proved unfavourable while en route to their intended winter base.
Polar scientific vessels therefore needed to be reinforced strongly enough to break through pack ice when conditions allowed, but also flexible to tolerate the forces imposed by an icefield if the ship found itself icebound. This state of imprisonment could last months at a time, and the ship needed to be fully seaworthy once released after a second or third winter. Many of the ships used for polar exploration had begun life as warships (reinforced against bombardment) or were refitted whalers or sealers. At the turn of the 20th century, when metal hulls had become the norm for other naval applications, wood remained a viable option for surviving the forces of pack ice without buckling: RRS Discovery, for example, was built with a multi-layered hull of overlapping oak, elm, pitch pine, and exotic hardwoods intended to maximise its strength.
Why bother with sails?
Discovery - the flagship of Scott and Shackleton's first Antarctic expedition - was one of the very last ships the British Navy ever built with masts, but they did have engines as well. The benefits of an engine were as obvious for the poles as anywhere else, if not moreso given the challenge of pushing through ice: they'd been used on British scientific expeditions to the poles since 1845 (Erebus and Terror, though their voyage ended unhappily.) There was, however, a pragmatic reason for this anachronistic choice: in short, polar explorers could expect to spend long periods sealed off from supply routes, unable to replenish their fuel stores while camped at the poles. Sailing when possible allowed vessels to conserve coal, and meant that they had a way home if something went wrong with the engine. After Endurance was lost, Shackleton's crew rowed the lifeboats to Elephant Island - but the subsequent ocean crossing to South Georgia was only possible under sail (with makeshift masts fitted to one of the lifeboats.)
When did they die out?
Rigged wooden vessels remained in use for polar research throughout the 1920s and into the 1930s, though primarily under steam power, and the next generation were made of metal and wholly reliant on engines, in an era of better communication and transport. Discovery remained in use for Antarctic research until 1931, when she was replaced by the next-gen Discovery II. Shackleton's last vessel Quest - wooden, rigged, but with a steam engine - carried polar scientific expeditions until WWII, then she was refitted with a diesel engine and remained in service as a sealer all the way into the 1960s.
Some sources:
I second what everyone else has said and have a few more things to add about the history of wooden-hulled, wind-propelled ships versus their metal-hulled cousins!
There's no one easy answer to this question because there was a good chunk of time where the two existed simultaneously. Steam ships were already in use prior to 1818. As stated here at the Royal Museums Greenwich, Isambard Kingdom Brunel's Great Britain was the first iron-hulled ship and propeller-driven ship (others had been the first to be either of those, but the Great Britain was the first to have both). It was launched in 1843 and was the first iron steamer to make a Transatlantic crossing. So, theoretically, you could say the slow and steady decline of the wooden, wind-propelled ship began between the 1820s and 1840s. But you could also say that it started even earlier with the first steam-powered boat Pyroscaphe in 1783, or, y'know, 5th century Greek mariners fitting out their ships with bronze sheeting. It's all a matter of opinion as to when the slow, slowwww decline of wood and canvas ships actually began.
From a polar exploration perspective like that of Shackleton and Endurance, the gruesome twosome of the HMS Erebus and HMS Terror are a good example of wooden-hulled but iron-clad ships driven by both wind and steam power, caught right at the precipice of the shift between the two. Both did Antarctic service under James Clark Ross and Francis R.M. Crozier, but when they returned to England and received their Arctic assignments, they were each equipped with a repurposed locomotive engine attached to that brand new screw propeller Brunel so kindly showed off only a few years earlier--as well as a hefty twelve days-worth of coal in storage. And while they weren't completely fitted out in iron, they did get some cladding on their fore and aft ends for reinforcement. These modifications were considered very high-tech for their age. (One Franklin academic at a conference described them as "space shuttles" of their age.)
Of course, those two are now very much so underwater like Endurance, but you get the idea. Advancements were being made left and right, but as u/LTNBFU pointed out from the Endurance book (which is VERY good), wooden ships were way cheaper. When you didn't have the glorious backing of a well-funded and bored-for-lack-of-war Admiralty the way Erebus and Terror had, it was much harder to foot the bill for a modded-out ship with all these newfangled features.
Wars have also given big ol' technological boosts along the way with regard to the way ships have been built and how quickly their modifications have been accepted as new standards. Obviously there are a ton of examples (we'd be here for a very long time if someone listed every ship with a metal hull or something other than wind power that appeared before the 1860s). The Monitor and Merrimack (technically the latter was the CSS Virginia made out of the corpse of the Merrimack, but whatevs) are great examples of ironclads getting thrust into the martial spotlight as early as 1862. One perfect quote from the link above states: "[The] battle demonstrated the viability of ironclad technology and provided a glimpse into the future of naval warfare."
So this takes us from 1818 to the 1860s with ironclad ships and all manner of new power sources, but wooden ships with canvas sails are still crisscrossing the seas the entire time and they're still, most emphatically, not dead yet. But the 1880s and 1890s introduced steel into the mix, and the cost issue of making wooden ships started weighing (heh) heavily in steel's favor over time.
Not to say it was a quick shift in the least, as evidenced by the Antarctic veteran and resident polar Tower of Babel the Belgica (speaking of great books, Madhouse at the End of the Earth is a wonder and a delight). Between 1897 and 1899, the crew of the Belgica, including a young Roald Amundsen, suffered in miserable conditions in a wooden, three-masted, iron-sheeted ship with a 35-horsepower steam-driven engine. Not that the engine did much other than take up space while the crew went frothing mad and ate penguin meat to fight off scurvy.
Anyway, to make a long story short, the final sustained death knell of the wooden wind-propelled ship probably came feasibly between the 1920s and 1940s. Welded steel and the introduction of diesel and gasoline as power sources helped the advancements along until wooden ships became museum pieces and novelties.
To answer the question in a solid, one-sentence way, though? Wooden, wind-propelled ships went out of favor some time between 1783 and 1940.
A sailing vessel would have been more appropriate for the voyage than steamships of the day. A Steamship on such a voyage would require numerous coaling coaling station stops, which didn’t yet exist along the route, as well as a much larger vessel to include an engineroom as well as sufficient space for coal, spare parts, additional crew, additional store. Commercial Sailing vessels remained in use well into the age of steam. I can’t recall exactly when, but they held the transatlantic crossing record for a long period.
Cheers
Capt. M.B. Miller