I don't think that "closed source / commercial alternatives" (IE in the browser case) have done any better on the bug side than the open source ones.So the received wisdom would have us believe. The thing is, my bug trackers objectively say different.
In the past 12 months, I have seen exactly one serious bug in IE (where I define "serious" to mean that user-visible functionality in at least one web app I've worked on does not work properly, even though the same functionality worked in earlier versions of the same browser and/or works in multiple other browsers). That one may have been IE's fault or it may have been a problem with a Java update, since the bad behaviour was at the interface between JS and Java and both had recently had updates when we started to see the problem.
In contrast, I have seen serious bugs in Chrome in every web app I work on, ranging from another Java integration issue through to messing up HTML5 video. Several of those bugs still haven't been fixed several update cycles later. Statistically, across all the projects I'm working on -- which since I'm doing various freelance work and running multiple businesses have little in common except for all being web apps -- it is clear that Chrome is by far the flakiest browser at the moment.
Firefox is actually faring slightly better lately, though as a Firefox user rather than a developer they still keep making usability screw-ups. I get that there have been a couple of critical security issues with Java recently, but since Firefox itself patches at least a handful of critical issues every release cycle I think the "Don't use this, run away, it's scary!" treatment is frankly rather childish. And as of version 18, Firefox frequently seems to corrupt the display of text areas like the one I'm typing this in.
I don't like to be negative about software since nothing's really perfect in this business, but I do point the finger squarely at the advocates of six-weekly releases on this one. The quality of Chrome has never been that great and it's always suffered from "UI rearranging itself at random" syndrome because of the silent updates. However, the quality of Firefox has nose-dived since the move to a rapid release cycle; I'd estimate we're seeing 4-5x as many minor issues in Firefox across all the projects I work on since that happened, plus a couple of howlers like the plug-in screw-up in the first so-called long term support release.