Starting with the newest form: B-pop, a style "invented" by a local music studio in Thimphu called M-Studio. They have produced several hits during their few years of existence, most famously Tharingsa of which I have heard at least four different versions (not to mention all the karaoke interpretations), and Yonphula which is probably their second-biggest hit.
The video below, filmed on a beautiful river-side with a prayer flag covered bridge in the background, is an acoustic version of singer-songwriter Pema Deki's "Due Atara". The faster, disco-type original song was chosen for the first ever flashmob organized in Bhutan: Pema Deki performed her song live when a group of 50 volunteers danced to it at the Clocktower Square in Thimphu on December 5, the International Volunteer Day 2013. Pretty great!
Rigsar is the name for Bhutanese popular music featuring synthesizers and elements from Western, Nepali and Hindi pop. Most of the song and dance numbers in Bhutanese films are of rigsar type, and the reason why people go to local cinema theatres at all. It is said that particularly the urban youth prefer rigsar - perhaps because it is easier to sing, it has a much faster rhythm than traditional Bhutanese music and the romantic lyrics usually circle around the typical boy-meets-girl scenario.
The three main instruments used in folk music are dramnyen (flute), chiwang (fiddle) and lingm (flute). There are many different singing styles, but here the focus is on the two most common ones: zhungdra and boedra.
The easier form of Bhutanese folk singing is boedra, which has evolved from Tibetan court music and has a rhythm. The more difficult style is zhungdra, traditional singing from the 17th century using "extended vocal tones in complex patterns which slowly decorate a relatively simple instrumental melody" (Wikipedia) and without a rhythm.
According to the Wikipedia article Music of Bhutan, "even those with natural singing ability, typically find it challenging to sing zhungdra". This is a mild under-statement: my spouse has lovely compared my attempts to sing traditional Bhutanese songs to a howling dog (so far). For a slightly more pleasant listening experience - and for seeing authentic village life in Bhutan - have a look at the video "Aum Nimchu Pem, legendary Bhutanese singer" produced by the Music of Bhutan Research Centre. Highly recommended!
If you want to find out more about the cultural importance of traditional music in Bhutan and see how dramnyen (lute) is played, check out this TedX Thimphu talk by the founder of the Music of Bhutan Research Centre, Sonam Dorji in November 2011.
And if you are really interested to learn more about the differences and history of both rigsar and traditional folk songs, an article by Sonam Kinga, "The Attributes and Values of Folk and Popular Songs" delves much deeper.
I will (perhaps) be able to sing some of the traditional songs more or less properly after a few months of practice. Until then I can only offer the best possible competition to all the wailing stray dogs on the streets of Thimphu...