I believe that links are not necessarily underlined. But unchecked links are crap. Sucks because an unticked link is a sign of a small site. But first things first.
Anyone who thinks at least a little about the usability (prastygospadi) of the site is well-known that the link should demonstrate with its appearance that it wants to be responsive to the user. To do this, there are three (known to me) ways:
1. Apply link as a button
The guys from Apple and their ilk are following this path. The menu on their website is the equivalent of the menu of any program in the Mac-axis. The method is absolutely nothing, but there is one problem: this method is suitable only for small sites with 2-3 pages. I can not imagine the site directory, where the link of links is decorated, like buttons, with arms, shadows and so on. The site Apple, I agree, rather big, so they use the second method to highlight links.
2. Highlight all links in one color.
It is also a good way, but often, well, it’s impossible to make all the links in one color. On the same Epple look - here and there links are blue, and somewhere completely gray on the contrary.

In addition, there is another big problem only the color identification of links. Even if you managed to maintain all the links in one color on your site (and for this, your site must be fresh-solid with a solid background), this means that you forgot about the excellent link property known as Visited, that is, the ability to change the style when visiting Usually the visited link changes color, automatically getting out of the general style of links. And for large non-linear sites, the visited links should be shown so as not to get lost.
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3. Finally, underline
This, in my opinion, is the most universal way to highlight a link. First, underlined links you can safely make out directories (dump sites), and secondly, you can without fear of using as many reference colors as your heart desires.
And combining these three ways, I think, is generally a mockery of the user. Instead of looking for the information he needs, he should understand the subtle idea of the designer.
It turns out that the first and second methods are not suitable for large sites, but on small sites it is quite acceptable. I am not considering web services, for example, Gmail, although this also sucks. Yandex.mail somehow got out, making dextpno-oriented service, but at the same time retaining the underlined links, for which they respect.
And everything would be fine if there were no customers who ordered portals and were firmly convinced that it was not necessary to scribble. And some funny argument is given: underlining looks like garbage :) They should write pictures, but not sites to do.