International Phonetic Alphabet (IPA) fonts
The following is a comprehensive list of modern open-source fonts that support IPA symbols.
IPA fonts with the best coverage are Charis, Gentium Plus, Doulus, and Andika, which were designed specifically for linguists by SIL International; as well as Google's Noto Sans and Noto Serif, who have the goal of covering as many Unicode symbols as they can in a neat, modern style (noting that no font in existence covers all 136,690 Unicode characters).
The other fonts on this list will cover all the IPA symbols and diacritics, however some have a few limitations to be aware of: some render some of the less common diacritics in slightly skewed positions, especially when attaching to symbols of differing heights; some may not render advanced tone contours (the ones that dyamically change in shape as you combine them); and some do not support ⱱ (which was a later addition to the IPA) or ᶑ (which is not officially recognised by the International Phonetic Association due to linguists not agreeing it exists). In any case, the problematic symbols are pretty rare.
Various Microsoft fonts also support IPA symbols extensively, but most are fairly dated, or too well-known to be interesting (with the exception of perhaps Cambria, which I quite like). These include Arial, Calibri, Cambria, Consolas (monospaced), Courier New (monospaced), Lucida Sans Unicode, Segoe UI, Tahoma and Times New Roman.
The fonts in no particular order:
Download ZIPName: Noto Sans
Released: 2013
Developer: Google
Name: Noto Serif
Released: 2013
Developer: Google
Name: Charis SIL
Released: 2006
Developer: SIL International
Name: Andika
Released: 2008
Developer: SIL International
Name: Gentium Plus
Released: 2001
Developer: SIL International
Name: Doulos
Released: 2006
Developer: SIL International
Name: DejaVu Sans
Released: 2004
Developer: Bitstream Vera
Name: DejaVu Serif
Released: 2004
Developer: Bitstream Vera
Coverage: No contour tones
Name: Linux Libertine
Released: 2012
Developer: Libertine Open Fonts Project
Name: Linux Biolinum
Released: 2012
Developer: Libertine Open Fonts Project
Name: Fira Sans
Released: 2013
Developer: Mozilla Foundation
Name: Lato
Released: 2010
Developer: tyPoland
Name: Tinos
Released: 2013
Developer: Steve Matteson
Name: Arimo
Released: 2013
Developer: Steve Matteson
Name: Voces
Released: 2010
Developer: Ana Paula Megda and Pablo Ugerman
Name: Judson
Released: 2012
Developer: Daniel Johnson
Name: Cardo
Released: 2010
Developer: David Perry
Name: Istok Web
Released: 2008
Developer: Andrey V. Panov
Name: Brill
Released: 2014
Developer: Brill
Name: EB Garamond
Released: 2011
Developer: Georg Duffner
Name: Source Sans Pro
Released: 2011
Developer: Paul D. Hunt
Name: Cousine
Released: 2011
Developer: Steve Matteson
Name: Quivira
Released: Unknown
Developer: Alexander Lange
Name: LeedsUni
Released: 2009
Developer: Alec McAllister
Name: Dialekt Uni
Released: 2002
Developer: Lars Törnqvist
Name: Hindsight Unicode
Released: 2000
Developer: Starving-4 Entertainment
Name: STIX
Released: 2008
Developer: STIX Pub
Name: Thryomanes
Released: 1998
Developer: Herman Miller
Name: RomanCyrillic Std
Released: 2006
Developer: Sebastian Kempgen
Name: Junicode
Released: 2001
Developer: Peter S. Blake
Name: Arev Sans
Released: 2014
Developer: Bitstream Vera Sans