The Council of Vocations giving Equality 7-2521 his job of Street Sweeper. This is important because this is what Equality is going to do for the remainder of his life and he doesn't even get a say in what he actually wants to do which is become a Scholar.
Equality seeing Liberty 5-3000 for the first time and becoming fascinated with her. When Equality first saw her he was instantly mesmerized and as he was describing her he stated, "Their hair was golden as the sun; their hair flew in the wind, shining and wild, as if it defied men to restrain it. We stood still; for the first time did we know fear, and then pain. And we stood still that we might not spill this pain more precious than pleasure" (Rand 39).
Equality discovering electricity when he was experimenting on a frog with wires and various metals. This is important because Equality has now discovered a new source of energy that could potentially improve society forever if the House of Scholars will accept the idea and him.
When Liberty (The Golden One) gives Equality the nickname, "The Unconquered" and brings water for Equality on a hot day of work. This is important because it shows how much Liberty cares about Equality and wants him to be well/safe.
When Equality creates his box-light bulb contraption and gets it to light up. When Equality realizes the difference he can make in society with this invention he states, "Then we thought of the meaning of that which lay before us. We can give our brothers a new light, cleaner and brighter, than any they have ever known. There are no limits to its secrets and its might, and it can be made to grant us anything if we but choose to ask" (Rand 60).
When Equality gets caught for being absent from the City Theatre and gets sent to the Palace of Corrective Detention where he is beaten because he won't tell the Council of Home where he was. This is important because it's when Equality realizes how cruel society is to their people and he also realizes that his invention is more important than physical pain because it can help society in the long run.