Final answer:
Advancement of free jazz/avant-garde in the late '50s and early '60s was spurred by venues like the Five Spot Café, record labels such as ESP-Disk, and landmark albums like Ornette Coleman's 'The Shape of Jazz to Come' and John Coltrane's 'A Love Supreme'.
Step-by-step explanation:
The nightclubs, record labels, and albums that advanced the free jazz/avant-garde aesthetic during the late 1950s and early 1960s can primarily attribute their cultural impact to a few key venues and figures within the jazz community. Notable nightclubs such as the Five Spot Café in New York City were known for hosting legendary musicians like Ornette Coleman, who is often cited as a pioneering force in free jazz with his album The Shape of Jazz to Come (1959). Another significant record label was ESP-Disk, which supported various avant-garde jazz musicians, including Albert Ayler whose album Spiritual Unity (1964) was pivotal. Lastly, John Coltrane's classic album A Love Supreme (1965), released by Impulse! Records, also greatly contributed to the exploration of the avant-garde in jazz.