Given how gee-whiz smart our smartphones are, with integrated messaging from a variety of sources, and enough computing power in your pocket to waste more cycles on animating a menu than the entire computer power of NASA during the moon landings, isn't it silly how voicemail is still a relic from the 1980s? The whole interface is an almost-robotic voice saying "You have six new messages." Then you can listen to them in order, with no indication who made any of them, no integration with your missed calls list or your address book, and a clumsy interface that requires you to keep taking the phone away from your ear and then back.
It's all because of the server-side nature of voicemail. You can't let voicemails go through to your phone or computer and let them answer it because half the point of voicemail is to catch calls when your phone is off, or out of range. But imagine if we let that same reasoning hold us back on email systems. Email has to be handled by the server for the same reasons, and yet our email experience robustly integrates with our address books, other messaging streams, and the full interface and data management capabilities of our computers or phones.
That's all possible because of server/client communication protocols like POP3 and IMAP. The server handles getting the mail and holding it. The client handles organizing and managing it. There's support for using multiple clients for your email: let your phone and your computer each get the email and handle it to their strengths (portability, or better data management and better user interface, respectively). There's support for the "last ditch" interface when you don't have a client (webmail, generally, though I remember the days when it was PINE in a telnet session).
Voicemail should be like this. The current "You have six new messages. Press 1 to listen to your messages..." system should be the equivalent of the webmail interface your ISP provides: something you use when you can't get to your own machine, but need to check your messages anyway. The rest of the time, your phone and/or computer should connect to the server regularly and download your voicemails, then integrate them with what they already have. Your missed calls log should be updated to show an icon that tells if a voicemail went with it, and thus the voicemails should also be tied to your address book so you know who the voicemails were from before you open them. Voicemails and missed calls could be listed amongst your other messages if you want. You could choose to listen to the important ones first. Attach them as a voice recording file to your to-do items or documents. Set filters to highlight voicemails from important people or block ones from annoying ones. Delete them from the server from your device, or only locally. Respond to a voicemail with an email since it's tied to their address book entry. Feed the voicemail through voice recognition to get the text (if your device is powerful enough for that). Forward voicemails as email attachments. And so on.
There are a few systems, mostly high-end and limited to within a specific system (usually a landline system), that do some of this, and even the iPhone includes a fraction of this functionality (largely as part of being tied to a single provider); but there's no technological reason why it can't be integrated to voicemail as fundamentally as IMAP and POP3 are integrated to email, and supported across many clients and many servers. It's just a matter of a standard protocol being promulgated and then developed around.
If you had all that and someone else had to use voicemail the way we nearly all do today, you'd look at them like they were living in the Stone Age. And if you had to deal with email the way you deal with voicemail you probably wouldn't use email. In the age of the iPhone and Blackberry, why do we live with voicemail systems that haven't evolved in two decades?