Visual List of iOS Fonts?

Have you tried iOS Fonts?

This gives you a visual of all of the available fonts with the ability to enter your own string of text to see how it would look.

This doesn't appear to have been updated for iOS 7 however but I am unaware of any additional fonts which have been added since iOS 6.

This font list was made with Xcode 7 and iOS 9.

If you are setting one of these programmatically and need the font name, you can do the following:

print(UIFont.familyNames)
// ["Copperplate", "Heiti SC", "Iowan Old Style", "Kohinoor Telugu", "Thonburi", "Heiti TC", "Courier New", "Gill Sans", "Apple SD Gothic Neo", "Marker Felt", "Avenir Next Condensed", "Tamil Sangam MN", "Helvetica Neue", "Gurmukhi MN", "Times New Roman", "Georgia", "Apple Color Emoji", "Arial Rounded MT Bold", "Kailasa", "Kohinoor Devanagari", "Kohinoor Bangla", "Chalkboard SE", "Sinhala Sangam MN", "PingFang TC", "Gujarati Sangam MN", "Damascus", "Noteworthy", "Geeza Pro", "Avenir", "Academy Engraved LET", "Mishafi", "Futura", "Farah", "Kannada Sangam MN", "Arial Hebrew", "Arial", "Party LET", "Chalkduster", "Hoefler Text", "Optima", "Palatino", "Lao Sangam MN", "Malayalam Sangam MN", "Al Nile", "Bradley Hand", "PingFang HK", "Trebuchet MS", "Helvetica", "Courier", "Cochin", "Hiragino Mincho ProN", "Devanagari Sangam MN", "Oriya Sangam MN", "Snell Roundhand", "Zapf Dingbats", "Bodoni 72", "Verdana", "American Typewriter", "Avenir Next", "Baskerville", "Khmer Sangam MN", "Didot", "Savoye LET", "Bodoni Ornaments", "Symbol", "Menlo", "Bodoni 72 Smallcaps", "Papyrus", "Hiragino Sans", "PingFang SC", "Euphemia UCAS", "Telugu Sangam MN", "Bangla Sangam MN", "Zapfino", "Bodoni 72 Oldstyle"]

And if you need the name of a particular bold/italic/etc style, you can get it like this:

for familyName in UIFont.familyNames {
    print(UIFont.fontNames(forFamilyName: familyName))
}