Events are "on". They happen. People who go to events are "at" events.

This discussion also needs to include use of the word "in". Being "at" a show means being there but not a participant in the actual show. "I was at the Rick Martin concert last night." would typically mean as part of the audience, but could also mean you were working security or at a food stall, as examples. Being "in" a show means you were a participant. If you are Ricky Martin's guitarist then you would say "I was in the Ricky Martin concert last night." When "on" is used it refers to the mechanism which was used to broadcast the show. You might be seen on TV or heard on the radio. If the concert was televised and your mother saw it she would say "I know you were at the Ricky Martin concert because I saw you on the TV." It can be confusing because she might say "I saw you on the Ricky Martin show" which is actually a way of giving two pieces of information: "I saw you on TV"; and "it was during the Ricky Martin show". If she had seen you at the show because she was working there, then she would not use the word "on". She would just say "I saw you at the Ricky Martin concert". The only reason she would use the word "on" in that context is if she didn't see you in person but via some mechanism. "I know you were at Ricky Martin last night because I saw you on the big screen". or she might have seen you on a security camera, or on a VIP guest list.

Well, thin of it this way - if you are on the stage or in front of the camera, you are on the show. "I saw the president on the news last night." If you are in the audience you are at the show. "I saw your brother at the concert last night."

Performing on the music show(on TV) Performing in the music show(live event or TV) Performing at the music show(live event) Performing at the music show on TV

Hmm, I think either can work but I would rearrange the sentence to "If you have any talents, you can display them at/on the show." Judging from that sentence alone I would use "at." Examples: "I saw the new Star Wars movie "at" the theatre." "I enjoy watching my favorite show "on" TV." So I think that if this person was displaying their talents at an actual show that you would attend in person, you would use "at." If it was on a show that you'd see on TV, you'd use "on."