It seems like the P-51 Mustang and the other equivalents at the time used mainly mostly multiple machines guns, like the M2 browning. Even moving into the Korean War, it seems like the F-86 still uses M3 machine guns, with the MIG-15 having autocannons and regular machine guns. However, it seems like basically all modern fighters use one single fast firing Rotary cannon, F-18, F-16,F-15, using what seems to be the M61 cannon with a fast firing rate or 6000rpm, with only the most F-35 braking from this trend(with only the A version having a built in a 25mm cannon). Even the Russian jets like the Su-35 have one single fast firing 30mm cannon. On top of that, it seems like all the jets have basically a bare minimum amount of ammunition, with at best at most 3 seconds a firing time.
Most people I've talked to said that the decline of the dogfighting and the air to air gunnery was the missile, however, I've also heard some say that there was an overlap when missile weren't reliable enough to depend on, yet good enough to carry, with the gun still being used often. Around when did the countries of the world switch from multiple machine gun armaments' on their fighter jets, to one single fast firing cannon?
4 months and no one's answered this, so I'll give it a go:
The underlying assumption was that nuclear weapons would make conventional wars obsolete and that as a result, air combat maneuvering (ACM) was dead, overtaken by the interception mission, which required supersonic speed, high technology sensors, and missile weaponry.”
The most basic answer of when the shift occurred for the US is with the development of the F-4 phantom. In fact the shift was initially not from multiple guns to one but from multiple guns to zero. The F-4 had no internally mounted cannon which proved a significant problem in Vietnam, its main theater, as the ROE during Rolling Thunder which required the primarily gun armed Migs 17s to attack first before they could be engaged and/or visual identification, at which point they were inside the engagement envelope of missiles.
The large, heavy, unmaneuverable and purely missile-armed fighters were forced to engage older but more agile Migs. The matter was made worse by the fact that ACM training left the curriculum soon after the Korean War. Political-based limitations on Rules of Engagement (ROE) further hampered fighter pilots. Incidents in which damaged American aircraft returned to base with American missiles lodged in them as well as fears that foreign non-combatant aircraft might be downed generated the policy requiring an aircraft had to be identified visually before it could be fired upon. This ROE negated many of the intended advantages of the missile-armed fighters of the period with their emphasis on Beyond Visual Range (BVR) capability. The resulting engagements frequently devolved into lower, slower endgame battles American fighters without guns could not fight well.
The combat lessons from Vietnam led to the rise of an air force "fighter mafia" to challenge the "bomber mafia" school of thought that had been preeminent in the USAF post wwii:
a small group of authoritative individuals, later to be known as the "Fighter Mafia," oversaw the creation of a report that called for new fighter designs that harked back to pre- nuclear fighter characteristics-- agile with good pilot visibility and armament for close engagements as well as BVR combat.
Their influence led to some of the design concepts behind the F-16 and F-15 and the reintroduction of a cannon on fighter aircraft (along with a general emphasis on fighter aircraft as opposed to interceptor aircraft as the "nuclear air force" had been trending towards pre Vietnam.) so the move wasn't actually from multiple guns to one it was multiple - > 0 - > 1
Source: Rediscovering Air, Superiority: Vietnam, the F-X, and the 'Fighter Mafia'; Cunningham
The Rules of Defeat: the impact of aerial rules of engagement on USAF operations in North Vietnam, 1965-1968; Drake
Edit: also as a note the navy F-4s never mounted an internal cannon because it shifted the center of gravity too far forward for carrier operations. Only the USAF F-4E did