How do I listen to sample/preview in the iOS Music app?

That would be part of the iTunes app that comes bundled (pre-installed) on your iPhone.

It was answered above perhaps, but the answer also applies to IPad. Do not open the Music App and hope to listen to music samples, like the old days. When you attempt to search for a song and try to play it in the Music app, you will get an annoying and repetitive "Join Apple Music" pop up panel, where you will be invited to get the 3 months free offer. Get out of Apple Music App completely.

First go to IOS Settings, scroll down to music and switch "Show Apple Music" to off. Then go to the iTunes Store App (not iTunes), and on the bottom, click the music icon, bottom, left side. Then you'll be able to do a search, click on the song/album title you want to hear, and listen to snippets, like the old, much better Apple Music set up. They over-paid $3 billion to Dre and Iovine for Beats Music and to recoup their investment, they try to push people into signing up. They don't have help info or docs in IOS/iMusic to clarify this misleading and confusing new protocol, which is purposeful.

Desktop Apple set up is different, but IOS/iMusic is now a sales funnel for Apple Music sign ups. That's why you'll have a difficult time trying to get info from Apple help pages or forums at Apple on simply how to play short samples. Sucks.