So, which one do you use? Most WebSDR sites now support both, and they offer, just above the waterfall display, controls where you can choose which one to use, separately for audio and waterfall. The site tries to make the best guess: if your browser supports HTML5 WebAudio it will be used; otherwise, it will choose Java.
If only those on port 80 work, you're probably behind a firewall that only allows port 80 (and a few others). This is often the case in public WiFi networks and office networks.
If only those on ports other than 80 work, you're probably behind a web proxy server, which doesn't know how to handle the (non-standard) audio and waterfall data streams. The proxy server may also be in your internet service provider's network.
On Android devices, you can access these using the Firefox browser; Chrome and Opera may also work (but not Opera-Mini), and it seems in the latest Android update (4.4), even the built-in browser supports this.
On iPhones, iPads etc, you can use the built-in Safari browser, but only if you have iOS version 6 or later.
Other SDRs typically use a fast A/D converter and digital hardware to filter part of the spectrum; they are typically connected to the PC via USB or ethernet. Unfortunately, there is no standardization among the interfaces for these SDRs, which makes it hard for me to support them. A generic interface for them is being added, but for now they cannot be used.
The (in)famous RTL-SDR dongles (cheap VHF/UHF SDRs) are now supported. Note that these SDRs have a rather small dynamic range, so should only be used in situations where there are no very strong signals. If you want to use these, please check carefully that they are not being overloaded, e.g. by comparing their reception (using normal SDR software) to e.g. a normal non-SDR receiver; in particular, pay attention to weak signals: they should not disappear into the noise.
However, this size is not so critical at all. The really important thing to get good reception with a MiniWhip, is its placement and grounding. I wrote some articles about this here and here. The latter has a paragraph at the end specifically about the setup used at the Twente WebSDR.