Yes, toddlers do already feel embarrassment, and it appears that this feeling starts during the second year of life, following the development of self-awareness. In the past, it has for instance been investigated by studying blushing in young children.

Embarrassment is quite closely related to shyness, feeling self-conscious, and anxiety. In the case of your child, it is not strange to think they might feel put on the spot or shy about their singing. Specifically, this can be called exposure embarrassment, when a child feels put on the spot, and/or feels like they are observed by others. They can also feel like this when they are asked to perform in front of an audience, looking into a mirror while other people look at them, and when they receive many compliments. In a study on 22-month-old children, they showed overt signs of embarrassment in these situations.

The feelings of embarrassment of your child are different than those of older children and adults. The feeling of embarrassment with toddlers is much more fleeting and bound to a specific event. When children develop a theory of mind (the ability to understand and take into account another individual's mental state), this changes their shyness and embarrassment. Research shows that generally when children are around 7, they start feeling worried about how they are perceived by others, and can fear they look foolish. So in the case of your kid, it was probably just the momentary embarrassment of being put on the spot, but not because they might feel juvenile or cool.

A little later on, children also develop second-hand embarrassment, like when they feel embarrassed by their parents. This starts when children are about 9. And from my experience as an educator: no matter how cool you are, your kid will most likely be embarrassed of you at some point. I taught pupils who parents were very seen as cool people by practically everyone, and still their kids were embarrassed.

If you would like to read more about it, this chapter is an interesting read, and it is almost fully available on Google Books:

Crozier, W. R. (2010). Shyness and the development of embarrassment and the self-conscious emotions. In K. H. Rubin & R. J. Coplan (Eds.), The development of shyness and social withdrawal. (pp. 42-63) Guilford Press.

Yes, toddlers do already feel embarrassment, and it appears that this feeling starts during the second year of life, following the development of self-awareness. In the past, it has for instance been investigated by studying blushing in young children.

Embarrassment is quite closely related to shyness, feeling self-conscious, and anxiety. In the case of your child, it is not strange to think they might feel put on the spot or shy about their singing. Specifically, this can be called exposure embarrassment, when a child feels put on the spot, and/or feels like they are observed by others. They can also feel like this when they are asked to perform in front of an audience, looking into a mirror while other people look at them, and when they receive many compliments. In a study on 22-month-old children, they showed overt signs of embarrassment in these situations.

The feelings of embarrassment of your child are different than those of older children and adults. The feeling of embarrassment with toddlers is much more fleeting and bound to a specific event. When children develop a theory of mind (the ability to understand and take into account another individual's mental state), this changes their shyness and embarrassment. Research shows that generally when children are around 7, they start feeling worried about how they are perceived by others, and can fear they look foolish. So in the case of your kid, it was probably just the momentary embarrassment of being put on the spot, but not because they might feel juvenile or cool.

A little later on, children also develop second-hand embarrassment, like when they feel embarrassed by their parents. This starts when children are about 9. And from my experience as an educator: no matter how cool you are, your kid will most likely be embarrassed of you at some point. I taught pupils who parents were very seen as cool people by practically everyone, and still their kids were embarrassed.

If you would like to read more about it, this chapter is an interesting read, and I could find it almost in full on Google Books

Crozier, W. R. (2010). Shyness and the development of embarrassment and the self-conscious emotions. In K. H. Rubin & R. J. Coplan (Eds.), The development of shyness and social withdrawal. (pp. 42-63) Guilford Press.