What are the various themes of "The Singing Lesson" by Katherine Mansfield?

One of the most important themes of the story is that of despair. Indeed, we're introduced to the theme right from the very first line:

With despair—cold, sharp despair—buried deep in her heart like a wicked knife, Miss Meadows, in cap and gown and carrying a little baton, trod the cold corridors that led to the music hall.

The opening line provides a suitable introduction to the first part of the story. Miss Meadows has just read the fateful letter from Basil and is utterly heartbroken, yet she still goes about her work, albeit reluctantly. Her despair is contagious, spreading to the girls in her music class. She chooses a piece for them to sing that perfectly reflects how she feels at that particular moment:

Good Heavens, what could be more tragic than that lament! Every note was a sigh, a sob, a groan of awful mournfulness.

The theme of appearance is also one to consider. Miss Meadows is concerned about what other teachers at the school will think when word gets out that Basil has effectively broken off the engagement. There is a sense that this is in some way more important to Miss Meadows than Basil's love:

"But, my darling, if you love me," thought Miss Meadows, "I don't mind how much it is. Love me as little as you like."

There is more than a faint air of desperation about these words. Miss Meadows lives at a time when unmarried women of her age (thirty) are considered "on the shelf," almost as old maids. The perception of other people and society as a whole are everything to Miss Meadows. And this remains the case even when she receives a telegram from Basil imploring her to ignore his previous letter.