Being able to treat the pattern of someone's brain as software to be run on a computer, perhaps in parallel or at a large speedup, would have a huge impact, both socially and economically.  Robin Hanson thinks it is the most likely route to artificial intelligence.  Anders Sandberg and Nick Bostrom of the Future Of Humanity Institute created out a roadmap for whole brain emulation in 2008, which covers a huge amount of research in this direction, combined with some scale analysis of the difficulty of various tasks.

Because the human brain is so large, and we are so far from having the technical capacity to scan or emulate it, it's difficult to evaluate progress.  Some other organisms, however, have much smaller brains: the nematode C. elegans has only 302 cells in its entire nervous system.  It is extremely well studied and well understood, having gone through heavy use as a research animal for decades.  Since at least 1986 we've known the full neural connectivity of C. elegans, something that would take decades and a huge amount of work to get for humans.  At 302 neurons, simulation has been within our computational capacity for at least that long.  With 25 years to work on it, shouldn't we be able to 'upload' a nematode by now?

Reading through the research, there's been some work on modeling subsystems and components, but I only find three projects that have tried to integrate this research into a complete simulation: the University of Oregon's NemaSys (~1997), the Perfect C. elegans Project (~1998), and Hiroshima University's Virtual C. Elegans project (~2004).  The second two don't have web pages, but they did put out papers: [1], [2], [3].

Another way to look at this is to list the researchers who seem to have been involved with C. elegans emulation.  I find:

• Hiroaki Kitano, Sony [1]
• Shugo Hamahashi, Keio University [1]
• Sean Luke, University of Maryland [1]
• Michiyo Suzuki, Hiroshima University  [2][3]
• Takeshi Goto, Hiroshima Univeristy [2]
• Toshio Tsuji, Hiroshima Univeristy [2][3]
• Hisao Ohtake, Hiroshima Univeristy [2]
• Thomas Ferree, University of Oregon [4][5][6][7]
• Ben Marcotte, University of Oregon [5]
• Sean Lockery, University of Oregon [4][5][6][7]
• Thomas Morse, University of Oregon [4]
• Stephen Wicks, University of British Columbia [8]
• Chris Roehrig, University of British Columbia [8]
• Catharine Rankin, University of British Columbia [8]
• Angelo Cangelosi, Rome Instituite of Psychology [9]
• Domenico Parisi, Rome Instituite of Psychology [9]

This seems like a research area where you have multiple groups working at different universities, trying for a while, and then moving on.  None of the simulation projects have gotten very far: their emulations are not complete and have some pieces filled in by guesswork, genetic algorithms, or other artificial sources.  I was optimistic about finding successful simulation projects before I started trying to find one, but now that I haven't, my estimate of how hard whole brain emulation would be has gone up significantly.  While I wouldn't say whole brain emulation could never happen, this looks to me like it is a very long way out, probably hundreds of years.

Note: I later reorganized this into a blog post, incorporating some feed back from these comments.

Papers:

[1] The Perfect C. elegans Project: An Initial Report (1998)

[2] A Dynamic Body Model of the Nematode C. elegans With Neural Oscillators (2005)

[3] A model of motor control of the nematode C. elegans with neuronal circuits (2005)

[4] Robust spacial navigation in a robot inspired by C. elegans (1998)

[5] Neural network models of chemotaxis in the nematode C. elegans (1997)

[6] Chemotaxis control by linear recurrent networks (1998)

[7] Computational rules for chemotaxis in the nematode C. elegans (1999)

[8] A Dynamic Network Simulation of the Nematode Tap Withdrawl Circuit: Predictions Concerning Synaptic Function Using Behavioral Criteria (1996)

[9] A Neural Network Model of Caenorhabditis Elegans: The Circuit of Touch Sensitivity (1997)