Whereas The Voiceless can talk and presumably does off-screen a lot, this character remains quiet most of the time, often out of shyness. However, around a select few people, he'll all of a sudden begin talking quite a bit. This character may in fact speak on-screen often (The Voiceless will at most get a few lines onscreen, no matter how much they're stated to talk offscreen), if they happen to be a main character - a role that most Voiceless characters don't do well in. That's because if the story's focus is on the character, it will naturally have a lot of scenes in which the character is with the select few he'll talk with.
The name comes from "elective mutism," which was the old term for this condition in real life. The name was changed to "selective mutism" due to the implication that the person was choosing not to talk, when in fact, the usual case was the person getting so terrified that their vocal cords froze up and they were literally unable to.
Compare The Quiet One. Some examples may overlap between these two. Also compare Silent Bob, Cannot Talk to Women, and Elective Unintelligible.
When adding to this page, please think carefully about whether your example fits. This trope is not merely someone who doesn't talk often or who doesn't talk except for one line. What is important here is that this character has a select few characters whom he will speak to. If you cannot say who those specific characters are within your example, it is not this trope.
Anime and Manga
In The Melancholy of Haruhi Suzumiya, Yuki Nagato rarely speaks more than a sentence at a time to anyone but Kyon. Justified in that she's an alien with limited information on Earth life, and more importantly (compared to others like her), was specifically created without social tools.
In the early parts of Bunny Drop, Rin falls into this. At first, the only one who she opens up to is Daikichi, in part because he resembles his grandfather (who is also Rin's father), and in part because he's at first the only adult who seems at all concerned for her well-being. It's most strongly demonstrated when, after slowly but surely opening up to others, Daikichi takes Rin to see his mother and father over the new year's holiday - and Rin immediately clams up again, remembering the harsh treatment she had previously received.
One of the traditions binding the shrine maiden Touka in Shindere Shoujo To Kodoku Na Shinigami is that she is absolutely forbidden from speaking with any man regardless of the circumstances, up to and including her own brother. The sole exception is her husband, when she chooses to take one.
Peter in Jumanji, who talks to no one but his sister ever since their parents death by car accident. Once Alan gets out of the game and finds his parents are also dead, Peter starts talking to him as well.
Charles Wallace was this as a child in A Wrinkle in Time. By the time of the later books, he has grown out of it.
The young adult novel Robin's Country centers around Dummy/Richard's learning to speak. He is this for most of the book after Marion discovers he is not actually The Voiceless (he sometimes talks in his sleep).
Lewis from the book Lightland is highly selective about who he talks around. He won't speak to his mother, teachers, classmates, neighbors- only to his friend Lottie. When he finally speaks to someone besides her, It's a pretty dramatic scene.
Ben from the book The White Giraffe is less selective, but still only speaks o his parents, and his new friend Martine. the other students don't believe he can talk at all. In the second book, he opens up a bit more, and everyone is shocked that he talks.
In the first book of the Elenium, the foundling Flute doesn't speak at all, except possibly off-page to Sephrenia, the only other member of her race in the party. Partway through the second book, she apparently decides this is tiresome and reverts to her true personality, which is decisive and bossy.
In Tobacco Road, Pearl is willing to talk only with her mother Ada, and even then uses few words. Jeeter reflects that Ada herself used to not speak to him.
In Peter Pays Tribute, Matt has gone more than a year without talking. When he finally does, he starts out only talking to his close friend.
The mostly silent Conundrum, from The X-Files episode "Humbug," spoke only one line, to Mulder and Scully and in the presence of Dr. Blockhead, at the end of the episode. It was implied he was very selective about those he spoke to.
Supernatural: After witnessing his dad's death, the son refuses to even talk with his mom. Dean gets him to open up little by little and by the end of the episode, he's back to the little kid he was before the incident.
In the play (and movie) Reckless, Pooty is this. Due to a Snowball Lie, her husband thinks she's a deaf-mute. When she's alone or with someone who will keep her secret, she talks.
On the Behind The Veil site there's Jennifer O'Connell, who stopped talking completely after the car crash that killed her parents and left her stranded in there for hours (due to her own paralysis). She only very recently started to talk again, and only to her step-mother or when the situation is important enough.