In combat with the enemy ofcorse. I always read about carriers getting hit by enemy planes or submarines. But I can't recall any instance of an enemy surface fleet getting an opportunity to directly engage a carrier in traditional naval warfare.
There are two well-known cases where carriers were engaged and sunk by surface ships. One was the Battle off Samar, part of the Battle of Leyte Gulf in October 1944, when a Japanese fleet engaged a USN task force of escort carriers. The other, which I'll describe in more detail here, was the sinking of the British carrier Glorious in June 1940.
Glorious had been built as one of the Royal Navy's 'large light cruisers' in 1916, but following the example of her half sister Furious, was converted to a carrier in the latter half of the 1920s. In the 1930s, she was used for training and exercises with the Mediterranean Fleet helping to develop the Royal Navy's carrier doctrine. During the Abyssinian Crisis, she was a major part of the RN's deterrent against Italian action. At the start of the war, she was still in the Mediterranean. In October 1939, she moved to the Indian Ocean to search for possible German surface raiders, but returned to the Mediterranean at the start of 1940.
Following the German invasion of Norway in April 1940, she was sent to join Home Fleet, along with Ark Royal. In late April, she ferried RAF Gladiator fighters to operate from a frozen lake in Central Norway. She also provided fighter cover for shipping into the ports of Namsos and Aandalsnes. At the start of May, she provided a brief cover for the evacuation of Aandalsnes, before returning to the UK. There, she loaded more RAF fighters - this time, Hurricanes of 46 Squadron. These were flown off, ultimately to Bardufoss, on the 26th May. She spent the next few days providing fighter cover, before returning to Scapa Flow to refuel on the 30th.
By the time she was ready to head out again, the situation had changed. The Allies were no longer willing to fight for Norway. Instead, they began to withdraw. Glorious became part of Operation Alphabet, the British evacuation from Narvik. Her role was to pick up RAF aircraft and return them to the UK. To allow space for this, her air group was reduced to just two squadrons of Naval aircraft. She arrived off Narvik on the 2nd June 1940, where she initially helped to provide air cover for the loading of troops. When this was complete, on the 7th, the RAF fighters were flown aboard. Ten Gladiators of 264 Squadron and 7 of 46 Squadron's Hurricanes landed on, the latter becoming the first high performance aircraft to land on a British carrier.
Meanwhile, there was a degree of discontent aboard. Her captain and the commander of her air group (the Commander (Air)) had fallen out significantly. This evolved to the point where her captain, Guy D'Oyly-Hughes, felt it was necessary to arrest the Commander (Air) and leave him at Scapa Flow before Glorious left for Norway. With the loading of the RAF aircraft complete, D'Oyly-Hughes wished to make a rapid return to the UK and bring this awkward situation to an end. In the early hours of the 8th June, he sent a request to his superiors, asking permission to return to Scapa Flow 'for the purpose of making preparations for impending courts martial'. This was granted - the threat to ships in the North Sea was felt to be low, as German surface vessels had not yet sunk a ship on the routes between Narvik and the UK. With this permission granted, Glorious left Narvik at 3:50 AM, escorted only by two destroyers, Acasta and Ardent.
Unfortunately, the Germans were at sea in force. A strong German force, consisting of the capital ships Scharnhorst and Gneisenau, the heavy cruiser Hipper and four destroyers left Kiel on the 4th June. This force, commanded by Admiral Marschall, was intended to attack Allied shipping off Harstad on the night of the 8th-9th. German intel on Allied movements was lacking, but on the 7th, Marschall received reports from aircraft that two groups of shipping were at sea. He chose to intercept the southernmost of these groups. This was a convoy of four ships - the troopship Orama, tanker Oil Pioneer and the hospital ship Atlantis, all escorted by the trawler Jupiter - edit: there was a mistake in the source I was using; the trawler was actually the Juniper. Marschall's force intercepted the convoy on the morning of the 8th, leaving the hospital ship alone but sinking the remaining ships. The presence of this convoy indicated to the Germans that an evacuation might be underway. Marschall detached Hipper and the destroyers to refuel at Trondheim and search for more convoys later, while Scharnhorst and Gneisenau continued towards the Norwegian coast.
At around 16:00, they sighted smoke on the horizon. This was Glorious and her meagre escort. She was not flying recon patrols that might have sighted the incoming German ships. Lacking her Commander (Air), with a captain who as a former submariner did not fully understand the capabilities of his aircraft, and with her decks and hangar choked by RAF fighters with no folding wings, this was understandable. Less understandable, no lookout had been posted in her crowd nest, despite conditions of extreme visibility. There had been reports of German activity on a radio channel for ships around Narvik, but Glorious was not listening in on this channel, having switched to a Home Fleet one when departing. This lack of awareness allowed the Germans to get into the best tactical position possible before attacking - they approached from windward, so that Glorious would have to turn towards them to launch any of her aircraft.
As soon as the British sighted the German ships, Glorious made smoke and turned away, while the two destroyers closed in to lay a smoke screen and to hopefully keep the two much heavier German ships away. Attempts were made to launch a strike against them, but no aircraft were armed and ready to fly. Her crew began preparing three Swordfish torpedo bombers, but their attempts were hampered by congestion in the hangars due to the RAF aircraft. A radio message calling for help was sent, but only on a low-power radio system that no other British warships received. At 16:30, the German ships opened fire at extreme range.
Hits were quickly scored on Glorious. An early hit knocked out her bridge, killing everyone there. Later ones caused heavy damage in her hangar. By 17:20, she was dead in the water. The order went out to abandon ship. With Glorious sinking, the Germans turned their attention towards the destroyers, which had closed to torpedo range. Ardent suffered several hits, sinking at about 17:28. Glorious followed shortly afterwards, rolling over and sinking. Acasta, the last survivor, managed to score a torpedo hit on Scharnhorst that caused serious damage, but was herself sunk at 18:06.
With the British force utterly destroyed, the Germans made no attempt to pick up survivors, instead heading north in search of further targets. While many men from all three ships had made it into the water, help took about 60 hours to arrive. Many of Glorious' life rafts were missing the food and water rations they were supposed to carry. This, combined with the cold waters of the North Sea, meant that only 46 men from all three ships were picked up. On the 11th, a Norwegian fishing boat picked up thirty-eight men from Glorious and one from Acasta, dropping them off in the Faroe Islands where they returned to British service. Another fishing boat picked up five men from Glorious, taking them back to Norway where they became German prisoners. Two from Ardent were picked up by a German seaplane. The remainder of the crews, over 1,500 men, died either aboard the ships or in the water awaiting rescue.
There were, in fact, two instances I know of where carriers were engaged by surface combatants. Samar in the Battle of Leyte Gulf and the Glorious off Norway. u/thefourthmaninaboat gives an excellent writeup on that so I'll cede the floor to them on that one.
BLUF: The US Navy made a blunder in the battle of Leyte Gulf and left a small group of carriers and destroyers alone against a large group of Japanese surface combatants. 1 carrier was sunk by direct engagement and another by kamikaze. Most other carriers were damaged and a bulk of the destroyers were sunk. To understand how this happened, for those interested, we'll dive into general naval warfare doctrine in WW2 and the battle itself.
The USN and IJN did everything they could to distance their carriers from any hostile gunnery. The Pacific War was very much an aircraft-centric battle where battleships themselves were mostly relegated to being a general threat and shore bombardment, particularly late in the war. Generally, the battleships were too slow to even reach US carriers unless something remarkable happened. Carriers, particularly the Fleet ones, were worth their weight in gold and both sides did everything possible to screen and protect them. The US had 3 different classifications for carriers (CV/Fleet, CVL/Light and CVE/Escort) with the rest being hastily designed and converted hulls of tankers, merchantmen and cruisers. The US built 150 carriers during the war with most of them being escorts which were slow, lightly armed and with a small complement of aircraft.
The US lost no modern (Essex class) Fleet carriers but did lose multiple escort carriers, mostly to kamikaze attacks as they generally were less defended and nowhere near as capable of withstanding a hit as the Essex class carriers were.
NOTE: The CVEs were lovingly called "Combustible Vulnerable Expendable" by sailors in WW2 to underscore how slow, vulnerable and high-priority targets they would have been seen as by the Japanese.
Leyte Gulf is a battle in late October (23-26) that actually is comprised of 4 major engagements and 2 smaller engagements, with the 4 major engagements able to be considered major battles in their own right. The Battle off Samar would be the 3rd of those major engagements and the build-up as to why it happened must be understood within the framework of the larger battle itself.
The Japanese strategy at Leyte was to lure, if possible, the overwhelmingly powerful US fleet away from supporting the landings at Leyte in the Philippines (which started on the 16th) while allowing their "Central Fleet" consisting of 5 battleships (including both the Yamato and Musashi), 12 cruisers and 15 destroyers to wreck havoc against US landings at Leyte, similar to their strategy 2 years prior at Guadalcanal. This fleet was sighted early on the 23rd and were shadowed by 2 US submarines. The US subs were able to sink 2 Japanese cruisers and forced 1 cruiser to pull back due to damage.
The first real engagement came on October 24 where US air elements engaged the Central Japanese fleet and sank the Musashi, the Yamatos sister ship, and crippled a cruiser. Otherwise, the rest of the fleet was able to progress onwards. The Japanese feigned a retreat and the US lost contact with them, assuming they had pulled back and away from Leyte.The next bit is the battle of Surigao Straight on October 25 where 2 Japanese battleships escorted by 4 destroyers were sighted and intercepted by 5 US battleships.
The cool part about this is 4 of the US battleships were sunk at Pearl Harbor and repaired and 3 of them were fully modernized with radar and fire control systems. This was also the last battleship-to-battleship engagement in history. The Japanese ships were no match for the modernized US ships as they were able to accurately fire and hit from extreme ranges of 13 miles/21km. For reference the Hood/Bismarck engagement was at 9 miles/14km.
At this point, Halsey felt fairly confident in his position. The main Japanese force had been engaged twice and had lost 1 battleship and 2 cruisers, with 2 more cruisers heavily damaged. The main force looked to have retreated from San Bernadino and the subordinate Japanese force had been utterly annihilated at Surigao so Halsey assumed the 2 entry points (San Bernadino and Surigao) were both soundly defended.
Halsey received reports of a large "Northern Fleet" of carriers and decided to advance to engage. This is where things got murky. Halsey took with him all available fleet carrier assets (CVs and CVLs) as well as a very powerful Battleship Task Force of 6 modern battleships to engage. This, again, followed doctrine as the battleships had a crucial role in air defense for the fleet carriers.
This left, as intended, the San Bernadino straight undefended. The Japanese were able to slip their still very potent Central Fleet (4 battleships, 8 cruisers and 15 destroyers) through undetected and engaged a reserve force of 6 escort carriers (again, these were slow ships at 18-19kt against the fast Japanese cruisers which were capable of 30-35kt) and 7 destroyers of which 4 were "destroyer escorts" and, when compared with fleet destroyers were similar to the fleet/escort dichotomy of the carriers. The 6 carriers, while they carried a decent compliment of aircraft, were mostly armed for air support and mostly carried high explosive and not armor piercing bombs.
The destroyers scrambled to engage the Japanese ships and the carriers launched strikes in rapid succession. The Gambier Bay, which was already slow, was caught by Japanese cruisers and sunk. The St Lo was the first US ship to be hit by and destroyed by a kamikaze.
The 4 US destroyers that sacrificed themselves to save the carriers effectively repulsed the Japanese advance while inflicting relatively high damage. Their exploits are legendary within US Naval history.
All in all, Leyte Gulf showed the obvious impact of radar and technological superiority of the US forces wherein destroyers were able to go to-to-toe with much larger cruisers due to accuracy of fire control and archaic battleships that had been previously sunk were able to land accurate fire from extreme distances. Additionally, it also displayed the impact of air power where the US was so eager to close with and destroy a Japanese surface fleet of carriers and negligently ignored a power carrier-less fleet... as well as the power of a (relatively) small compliment of aircraft against a vastly superior on paper fleet where US aircraft effectively sank 2 cruisers (the other was sunk by 5" shells) and heavily damaged most other ships in the fleet save for the Yamato.