A very common issue when using CSS3 to add border-radius to web site elements is incompatibility with older browser. To find out which browser is compatible with this cool feature please visit http://www.css3.info/preview/rounded-border/ I have included and extract below.
Once you have determined compatibility the next question you should ask yourself is “What is the market share of each browser?” Fortunately the nice people at www.netmarketshare.com answer that question http://www.netmarketshare.com/browser-market-share.aspx?qprid=2&qpcustomd=0
The biggest culprit for lack of compatibility with CSS 3 is Internet Explorer 8 which at the time of writing has 20% market share of the desktop market.
At present Opera (version 10.5 onward), Safari (version 5 onward) and Chrome (version 5 onward) all support the individual border-*-radius properties and the border-radius shorthand property as natively defined in the current W3C Specification (although there are still outstanding bugs on issues such as border style transitions, using percentages for lengths, etc.).
Mozilla Firefox (version 1.0 onward) supports border-radius with the -moz- prefix, although there are some discrepancies between the Mozilla implementation and the current W3C specification (see below).
Update:Recent Firefox nightly versions support border-radius without the -moz- prefix.
Safari and Chrome (and other webkit based browsers) have supported border-radius with the -webkit- prefix since version 3 (no longer needed from version 5 onward), although again with some discrepancies from the current specification (see this article for further details of how older versions of Webkit handle border-radius).
Even Microsoft have promised, and demonstrated in their recent preview release, support for border-radius from Internet Explorer 9 onward (without prefix).