Is it possible to trigger share menu on smartphones (via HTML/JS)?

It is possible with a big catch. Currently only available in Chrome for Android, Samsung internet and on Safari (desktop and mobile). And support is coming to Edge and Chrome on desktop http://caniuse.com/#feat=web-share

if (navigator.share) {
  navigator.share({
    title: document.title,
    text: "Hello World",
    url: window.location.href
  })
  .then(() => console.log('Successful share'))
  .catch(error => console.log('Error sharing:', error));
}

https://developers.google.com/web/updates/2016/10/navigator-share

I added this as all answers seems outdated by 2018-07-16.

It is possible, but only in a few browsers (MDN Reference), achieved througth the one method API in navigator:

navigator
    .share({
        title: document.title,
        text: 'Hello World',
        url: window.location.href
    })
    .then(() => console.log('Successful share! 🎉'))
    .catch(err => console.error(err));

Google's reference: https://developers.google.com/web/updates/2016/10/navigator-share

Also, there was a thing called Web Intends which is a dead project, you should go with navigator.share instead.