While it's generally best to run the latest versions of applications sometimes you can't because they just don't work. This can happen if you are using an old or low memory device that doesn't have enough resources or the right Java JSR to run the current versions. A while back I posted links to old versions of Gmail. To find those I had to do a lot of digging through old forum posts and trying different google.com URLs to find one that worked.
If you are looking for an old version of UCWEB or Opera Mini it's a lot easier as a couple of helpful users have compiled lists of direct download links on the Opera Mini and UCWEB user forums.
The UCWEB list is in a forum post by JBpseudo titled UCWEB Direct Download Links for All Final Versions, All Platforms. It contains a lot of gems like links to English versions of 64 KB and 128 KB low-memory versions of the browser. Often these low memory version will work on phones that have enough memory but lack JSRs needed by the full version. Some of these, like the UCWEB 6.0 64 KB version are unofficial translations and still contain some Chinese text on the home screen. But the menus of all seem to be in be in English.
I had high hopes for finding a version of UCWEB that would run on my BlackBerry 7100i where I get a Java error on startup with all the full release versions. Both the 128 KB files loaded and ran but were too slow to be usable. The 64 KB variant also ran but refused to load any URLs at all. My dreams of having UCWEB on the 'Berry have been dashed again. However, users on the forums have been reporting success with the low memory versions on many phones especially older Nokia S40 models, so give them a try if your phone can't run the official full version of UCWEB.
For most releases the page only has a the .jar file which is the actual executable. Some phones also need a .jad file which you can easily create on a PC using the free JadMaker program.
For old versions of Opera Mini visit Serola's post, Opera Mini direct download links (new) on the Opera Mini Forum. It has links to the generic Java, Palm and BlackBerry versions of almost every Beta and Release of Opera Mini going back as far as 2.0. If you are having trouble getting the latest Opera Mini to work try, some of the earlier ones especially the "Basic" version of 3.1 which is compatible with most phones.
Even if the current Opera Mini works on your phone you might want to install 3.1 Advanced and one of the 4.0 or 4.1 Betas. Copies of 3.1 Final, 4.1 Final and one Beta can coexist on most phones. BlackBerries are limited to a 4.n Final OR 3.1 Final plus one Beta.) If you have a phone that can run multiple java apps simultaneousl, like most Sony Ericsson, S60 and Motorola iDen models you can use multiple installations as a workaround for Opera Mini's lack of tabbed browsing by launching a couple of instances of Opera Mini and loading a different site in each.
The 1st 4.0 Beta is especially interesting as it seems to be the only Opera Mini 4.n version that can load all the images on big pages like the Wikipedia USA entry. Supposedly large pages were crashing this version on some phones so Opera added code to later releases that limits page size by not loading images on pages over a certain size. However I haven't had any problem using Beta 1 to load huge pages on my N95 and Motorola i850.
Another reason to install Opera Mini 3.1 is that it uses a different architecture and server farm than the 4.n serries and will sometimes load pages the current release can't.
If you're a fan of the fit-to-width "mobile view" in Opera Mini 4 like I am you might actually prefer 3.1 for general browsing. Mobile view in 4.n has a couple of the annoying bugs that aren't present in 3.1. In the current version small images like the stars I use for rating sites on ths blog are not displayed in mobile view. Another problem with 4.n is that on many sites, including all Blogger blogs, mobile view flat out doesn't work and you have to scroll horizontally to see all the content. Of course, you give up a lot with 3.1 including Opera Link and the ability to save pages and download and I end up of using 4.2 most of the time. Still it's good to have choices.