In the Series
Simon was first seen at the beach with his father and 5 brothers, reading in one of his father's old wizard books. When shooed away from the beach, he asked about wizards being the best. He was an ambitious and competitive boy who dreamed about becoming an ExtraOrdinary Wizard. When the Heaps fleed from the Supreme Custodian, Simon sneaked away to find his sweetheart, Lucy Gringe, and attempted to marry her but failed and got locked away. He would occasionally talk to the Supreme Custodian trying to get information and giving him ideas about Darke Magyk. He escaped and reunited with his parents, brother, and sister just in time for Boy 412's Apprentice Supper, and the revealing of Boy 412's true identity-Septimus Heap. Angry with the boy, saying that Boy 412 was not Septimus and for not getting the Apprenticeship that he wanted, Simon ran off and found the bones of DomDaniel, and revived him and became his new Apprentice. 
Simon kidnapped Jenna and wanted to kill her. She escaped him and he set his Tracker Ball named Sleuth after her. He attacked Jenna, Septimus, Nicko, and Wolf Boy on the Causeway and showed off his new gift of Flyte to them. Simon wal also involved with a Darke plan to bring back DomDaniel. He put a bone of DomDaniel in every piece of a ShadowSafe built to get rid of Marcia's troublesome Shadow. Marcia was able to survive, and Septimus got Simon's piece of the Flyte Charm after he attempted to attack Septimus and Jenna while they were on Spit Fyre. Septimus said Simon from a huge fall, and reunited his Flyte Charm with the one Marcia gave him when she asked him to be her Apprentice. In the end, Simon leaves the Castle. 
Later, Simon wished he didn't get involved in Darke Magyk. He kept trying to help Marcia or give her information, hoping that she would forgive him.
Simon helped Septimus figure out where the entrance to the Darke Halls was, and gave him advice on how to get out. He also explained to Marcia how to get rid of the Darke Domain. 
Before the events of the TodHunter Moon Series, Simon and Lucy have a son together, William.
A blind transport infected Simon's right eye with Lapis Lazuli, filling the entire eye socket with the magical blue stone, and possibly extending into his brain.  The right eye was able to be infected due to an existing scar, which he received from Wolf Boy's slingshot.
Simon was very powerful Wizard. Though not quite as gifted as Septimus , he far surpassed the remainder of his family including his father, Silas Heap . After studying with DomDaniel , he also became a gifted user of Darke Magyk , becoming a Darke Wizard until he quit using Darkenesse in later books when he "turned Lyte." Despite having renounced the Darke side, some of his involuntary Darke gifts, such as the art of Suspension, can still kick in and have proved to be very handy. Simon's knowledge of the Darke becomes key when he and his family are confronted with the Darke Domaine, although that same knowledge was the very reason the Domaine was able to attack the Castle at all, as he was forced to open the door and bring down the Safety Curtain around the Palace.
Most notably, Simon could cast a very powerful ThunderFlash which he demonstrated on Hotep Ra's Dragon Boat, placing it in a comatose statement after the effect. He can also summon FireSnakes to attack and induce Darke Sleepe in even animals as large as horses. Simon also EnGendered Sleuth and made it dangerously effective, able to track Jenna all the way from the Badlands to the Port despite several days' head start. Simon removes all traces of Darkenesse from Sleuth after the events of Flyte, which Marcia discovers when she comes to ask Simon to track Septimus.
Simon also became fairly good with the power of Flyte Charm however he never mastered it like Septimus because he lacked the complete Charm long enough to do so. He was able to fly very high and attack his enemies from the air. 
He also owned the Camera Obscura, after DomDaniel passed away at last, which he used for spying on the peoples of The Castle during his Darke times.
As noted by Septimus, Simon possessed the same level of proficiency as Septimus himself, despite being largely self taught most of his more advanced skills and his younger brother's powers as a seventh son of a seventh son. 
During the events of Fyre, Simon is taken on as Marcellus' Alchemie Apprentice, to replace Septimus. While Simon is not seen doing much Alchemie during the story, instead helping Marcellus reconstruct the Great Chamber of Fyre, Marcellus and Septimus both note his aptitude. Simon's knowledge of the Darke makes him a proficient Alchemist, for the Alchemists have been closer to the Darke Side than most Wizards get.
Simon is a skilled horseman, mastering Thunder (although Lucy notes that even Simon is wary of the horse's occasional tendency to kick. Like his master, Thunder has something of the Darke in him, and is able to travel through Darke Domaines, although Simon must sedate him when the concentrated Darke Domaine traps the horse in the Heaps' old apartment.