First, we have a few honorable mentions:
Okay, so here's the countdown:
#5: Youth of Today
This band invented youth crew, and was one of the greatest straight edge bands because they made being vegetarian a part of straight edge with the song "no more". YoT were one of the important pioneering straight edge bands.
Essential album: Break down the Walls
#4: SSD (Social System Decontrol)
SSD were a band from Boston who were short lived, but (like fellow pioneers Minor Threat) a very important band to the straight edge world. They had such intense, heavy songs and were one of the most angry bands ever to be on the scene.
Essential album: The Kids will have their say
#3: Gorilla Biscuits
Gorilla Biscuits (sometimes abbreviated to GB) were another youth crew band, but had such a different sound from every other NYHC (New York hardcore)band, because with GB, it wasn't just sheer brute force that made them great, it was also their intelligence.
Essential album: Start Today
#2: 7 Seconds
Also one of the founding melodic hardcore bands,7 Seconds had an awesome force that combined both melody and speed. They had such an energy that no other band in the punk scene has captured and they have so many great things about themselves.
Essential album: Walk Together, Rock Together
#1: Minor Threat
Minor Threat were the first straight edge band. Yes, the teen idles kind of embodied the straight edge attitude, but Minor threat had great lyrics, great riffs, and amazing vocals from the god of straight edge: Ian McKaye. No one can say that they are straight edge without at least owning Complete Discography. Minor Threat demolishes all other straight edge bands.
Essential album: Complete Discography