Describe opera in the twentieth and twenty-first centuries. Give specific examples of different styles.

The first thing I would note, to begin with, is that the general taste of opera-goers tends to be very conservative, musically thinking. While there are new operas being performed, if we were to look at the statistics of the most popular and commonly performed operas across the world, we'd see classics dominating those statistics: Verdi, Mozart, Puccini, and the like. In a sense, contemporary opera tends to live in the shadow of the Carmen's of the world.
That being said, you do track your question back into the twentieth century, which means that one of these giants actually does apply to your question: Giacomo Puccini, who died in 1924. So works like Turandot and Madame Butterfly definitely would apply to this analysis, from the first quarter of the century.
Anyway, there are a few themes I would express. First of all, we do have new themes and evolutions within music in the twentieth century, and I would expect that certain operas would reflect these characteristics. Composers like Schoenberg and Stravinsky challenged musical aesthetics in radical ways, and they do have operas in their oeuvres, and I would expect their work to be more modernist approaches in classical music. Additionally, I could mention someone like Gershwin and Porgy and Bess, whose work contains very strong jazz elements.
The last thing I want to mention is musical theater, because I do think that the division between musical theater and opera tends to be a bit unclear. Theater composers such as Andrew Lloyd Webber can get extremely operatic in their approaches to music. I'd also mention a musical such as Les Misérables (music by Schönberg), which is performed completely sung through, utilizing leitmotifs to announce characters and plot developments. On the other hand, I could mention something like Mozart's Magic Flute, which alternates between speaking and song, much in the way we would now associate more in the manner of musical theater. At a certain point, I do feel like the division between the two can be put into question.