Short answer:
All of them.
Long answer:
As a web developer you won't get around testing and debugging your applications regularly with multiple browsers. No matter how much you like BrowserX and dislike BrowserY, when the bug only occurs in BrowserY, you need to debug it with the tools BrowserY provides. So as a web developer you need to relinquish the mindset of "your" browser and get used to using all of them simultaneously for development.
However, many browsers use the same rendering engine and only differ by the user interface around them. When a website works in one browser of the family, it usually works in all of them. So for testing during development (proper QA testing before release is a different issue), you need one browser of each family:
- Google's Blink: Chrome, Chromium, Opera (recent versions) and a large number of niche browsers all use the Blink rendering engine and the V8 Javascript engine.
- Apple's WebKit: Blink is based on WebKit, so they are still very similar. But both projects start to diverge, which means testing on both branches becomes more and more necessary. The most well-known browser which uses Webkit is Apple's Safari.
- Mozilla's Gecko: The most well-known user is Firefox, as well as any other Mozilla-based products.
- Microsoft's Trident/Edge: Used by Microsoft Internet Explorer, obviously, and more subtly by many other programs on Windows which display HTML content as a secondary function. MSIE was always a major headache for web developers due to Microsoft's update policy. Newer versions are only available for the newest Windows version at their release date, which means in contrary to other browsers, older versions retain considerable market share. To make matters worse, it is very tricky to install multiple versions of IE on the same operating system. But fortunately Microsoft has finally understood that developers are important to them, so the development tools for Internet Explorer 11 got an emulation mode for earlier versions down to the antique version 5. The emulation is quite reliable, so you only need IE11 to test the whole IE family. Using virtual machines with multiple versions is usually not necessary anymore.
The biggest problem with these is that while recent versions of Safari are only available on iOS, Internet Explorer 11 is only available on Windows 7 or later. So you either still need at least two operating systems or need to do without testing one of them. Microsoft conveniently provides free-of-cost virtual machines pre-configured with various combinations of Windows OS + Internet Explorer intended for testing web pages/apps.
