Singapore was the linchpin for British control in the Far East. As you have noted, it was referred to as the "Gibraltar of the East" because of how heavily fortified it was. Yet, it was overconfidence in the defences of Singapore that were to be one of the reasons for Japan's swift victory. Also, the British paid very little attention to the defence of the Far East, seeing it as less important compared to the war in Europe and Africa. The British command in the Far East had not received any where near enough aircraft and the aircraft they did receive were outdated. The famous "guns of Singapore", contrary to popular belief, they were able to turn inland, but they lacked the ammo necessary to fire on Infantry, the only ammo they had was armour piercing shells meant for sinking ships. In 1940 the Japanese had sent military officers into Singapore to observe the defences , and they correctly noted that Singapore could only be attacked from the North, from across the strait of Johore. The British command was also filled with indecisiveness, do they defend only Singapore, or do they defend the whole Malayan peninsula? Several officers had noted that Singapore wasn't the "fortress" that it was made out to be, and that Britain's survival would depend on defending the whole island, but despite a lot of talk, very little change in the defensive plans actually happened.
On December 4th, the Japanese invasion force embarked and headed towards Malaya, they were commanded by Yamashita Tomoyuki; a rising star in the Imperial Japanese Army. He was confident, even taking time to write poetry like this:
On the day the sun shines with the moon The arrow leaves the bow It carries my spirit towards the enemy With me are a hundred million souls My people of the East On this day when the moon shines And the sun both shine
He had every right to be confident, the Japanese would have complete aerial dominance. The Japanese had stationed 600 quality aircraft in Indochina, and these aircraft outclassed and outnumbered anything the British could them. The Japanese also had detailed knowledge of the British defensive situation and there British defensive plans. This was thanks to the "Automeden" incident. The Automeden was a British ship that got attacked by a German surface raider. The ship was carrying secret documents that gave details on the British defences in the Far East, and these were promptly handed over to the Japanese. This incident actually played a vital role in convincing the Japanese to go to war, as it showed how weak the Allied defences in Asia were.
At 1:30 AM on December 8th, the Japanese landed at a place called Kota Bahru. The Japanese quickly overran the defences and destoryed around 60 British planes in doing so. The British to make a quick retreat inland. As the British retreated they left behind motor vehicles and airfields, all of which the Japanese happily put to use. Around this time "Force Z", the British naval task force sent to help Singapore, left its temporary base in the Philippines and sailed towards Singapore. The admiral of the force refused air cover, and he payed for it as Japanese aircraft sunk destroyed his force, sinking one of the Royal Navy's most modern battleships, "The Prince of Wales".
The British soldiers were generally inexperienced and made a number of false assumptions. Firstly they underestimated the Japanese fighting ability, they assumed that Japanese soldiers could not tolerate the jungle conditions of Malaya, and finally they assumed that Japanese soldiers would be hampered by lack of infrastructure. The Japanese proved all these assumptions wrong. The Japanese troops in Malay were battle hardened and many of them had fought in China. The Japanese were adept at jungle warfare, they wore light clothing that made navigating through the jungle heat easy. This came back to bite the British, who's soldiers were not experienced in jungle warfare, many times the Japanese would simply go through the jungle and appear behind the British lines. And finally the Japanese used bicycles to great effect; this "bicycle blitzkrieg" was extremely effective as large amounts of Japanese soldiers were able to outpace the British and overrun the British defensive positions almost as soon as they were established.
After their initial landing the main Japanese advance turned to the West coast of Malaya. The British retreated back to a defensive line that, according to its commander, should have held out for three months. But the Japanese used their small number of tanks to crash through the line in fifteen hours. The Japanese were able to supply many of their needs using captured British supplies, which speaks to the speed of the British collapse. Next, the Japanese took out the fortress of Penang, which was one of the oldest British possessions on the peninsula. The Japanese taunted the British with radio broadcasts as the Japanese began to bomb Singapore:
you English gentlemen: “How do you like our bombing? Isn't it a better tonic than your whisky soda?”’
As the Japanese bombing campaign continued, the British administration began to fall apart. Desertions from local units became common, and morale plummeted. Despite "fortress Singapore" still standing, most people knew that the British couldn't hold out for much longer. The last obstacle for the Japanese on the road to Kuala Lumpur, was a rocky bastion near the town of Kampar. The 11th Indian division made a stand here, but was quickly pushed back by the Imperial Guards division of the Japanese Army, they fell back to the Slim River where the division was destroyed by the Japanese tanks. Despite all these setbacks, the arrival of British reinforcements from the Middle East meant that the British still outnumbered the Japanese.
The end came for Singapore on February 15th, when the reservoirs that supplied water to Singapore fell into Japanese hands. The Japanese had gotten across the strait of Johore and onto Singapore island on February 7th. They had beaten back the Australian 22nd and 27th brigades which had been given the task of keeping the Japanese off the island. Arthur Percival the commander of the British land forces in Malaya was told to hold Singapore until the last man, but he had a crisis on his hands. The population of Singapore had swollen to well over a million due to massive amounts of refugees, and he didn't have the supplies to keep them and his troops alive. Percival surrender to the Japanese along with over 100,000 Commonwealth soldiers. For his part Percival would hold some minor offices later in life, but the fall of Singapore would always be a stain on his record. The Japanese commander, Tomoyuki Yamashita, would later be put into a militarily insignificant post in Manchuria, he would later be called out of this exile to participate in the defence of the Philippines. He would be executed by the Americans at the end of the war for war crimes.
Ultimately the British defeat is a classic case of a smaller, well prepared force, defeating a larger, unprepared force. The British may have been able to put up a reasonable defence, but with their attention focused on Africa and Europe, the Far Eastern theatre kinda got lost in all the commotion, and the people of Malaya and Singapore suffered for it.
Books I would recommend for this topic are "Forgotten Armies" by Tim Harper, which is one of the best books about the British campaign in the Far East. Also, for an overview of the Japanese Army, look for "Japan's Imperial Army: Its rise and fall" by Edward Drea. For a political look at effects of the British surrender at Singapore, I would recommend the book "The Rise and Fall of the British Empire" by Lawrence James.
The Japanese had massive air (4-1) and armor (400-0) superiority. And they had battle experienced troops. The British troops in comparison had very few men with any combat experience. And even though they were theoretically outnumbered, the Japanese usually had equal numbers of troops in individual engagements. The British were spread over the entire peninsula.
With their tank and air advantage the Japanese were able to use mobile warfare to repetitively outflank and outmaneuver their opponent, rapidly pushing them down the peninsula.
Even once they were just defending Singapore the British didn't have enough troops to defend the entire island. And all the fortifications were set to defend from a seaward assault, not an assault from the mainland. They could turn and fire north. But they were ill positioned and had mostly AP rounds as they were intended to to defend against battleships. So the Japanese were able to identify a less defended point and forced a crossing. Once across the Japanese were able to capture the main water supply for the island, and the primary food depot.
In all the Japanese brought a smaller but much better equipped, trained, and led army to the fight.
I actually did my undergrad thesis on the fortress at Singapore. In addition to what others have said, one of the key things about the development of Singapore was that it was constantly undersourced. Thus, the island didn't have all the defenses and bases that were originally envisioned. This has to do with the fact that the public was in no mood to increase defense expeditures after spending so much during World War I. So, the British military didn't really have the budget they needed to make Singapore truly defensible.
However, the most important thing about to know about why Singapore fell is because British strategy for defending the Far East was predicated on shifting significant resources from the Atlantic, the Med, and Africa to Singapore. Thus, if war were to be declared, the British fleet was supposed to transfer from the Atlantic to the Pacific with as many troops, planes, and supplies as possible. World War II threw this plan out the window because Britain's number one priority was fighiting Germany and Italy. When Japan declared war on Britain, Britain had no fleet, armies, or squadrons of planes to spare. The Japanese thus had superiority in everything that mattered and by wide margins. Realistically, there was just no way Britain could defend Singapore with the resources they had even if the defenses at Singapore had been fully built.
There are a couple of reasons.
First one was that the Japanese had a much higher morale. Many troops defending Singapore at the time were just sent over from Australia or England who didn't consider Singapore their home and didn't want to fight to defend it. Whereas the Japanese were fiercely devoted to their emporer and the war. Of the 85,000 soldiers defending singapore 80,000 surrendered and just 5,000 were killed or wounded.
The 2nd one is that the Japanese had air and naval superiority. Their zero planes were really good at the time and were much better than whatever the defenders had. And the defenders had pretty much no control of the seas as their only 2 battleships had already been sunk.
Last one would be that the Japanese had better tactics and better trained troops in general. They managed to trick the defenders by sending a small force to the north east of the island while actually focusing their main force to the north west side which made the defenders move their soldiers away from the west side to the east side. The Japanese troops were also able to move through the jungles of malaya and singapore easily on their bicycles as they had prior experience and training. Many of the defending soldiers were new recruits or poorly trained and some were even still taking shooting lessons while being bombed from above.
I have always heard of Corregidor as the "Gibraltar of the East". Is that an American versus British difference?