Where can I find a Bitcoin-related job? Say I want to use a Bitcoin expertise I have and turn that into a career. Is there any website listing job offers related to Bitcoin that I could use? Alternatively, if I am someone looking for Bitcoin-savvy people to employ, where should I start looking?
You are describing, the new cottage industry -- that of The Bitcoin Specialist.

There are a lot of skills that can be combined to become self employed providing Bitcoin-related services.

There is profit that can be made when trading bitcoins in-person.  There are hundreds of individuals who are signed up with LocalBitcoins and offer this personal buying and/or selling service.  Adding a fee or offering an exchange rate in your favor from five to fifteen percent versus the market rate is not uncommon, though that is temporary as competition will bring the rates more in line with market prices over the long run.


http://www.LocalBitcoins.com


There is the need for IT security assistance.  It could be as simple as helping a person understand two-factor authentication and setting up Google Authenticator to secure access to the user's exchange account.

Individuals who don't know what an "air gap" is might be willing to pay to be shown how to create an offline paper wallet using a LiveOS on a system that has no network connectivity -- purposely disabled for security purposes.  Or set up an Armory offline wallet and demonstrate how it is used:


http://www.BitAddress.org
http://bitcoinarmory.com/index.php/using-offline-wallets-in-armory


More and more Bitcoin-related services, including #bitcoin-otc marketplace, MPEx, and FastCash4Bitcoins.com for example, require GPG. Setting that up for people (and showing how to back up the GPG private key) is a needed support service.

Online there are people who simply want to move their funds from their own wallet to an EWallet and can't be bothered with updating the software and/or waiting for the blockchain to finish downloading.   But they might be willing to attach their wallet.dat (if encrypted) or upload it (via https) to someone offering wallet transfer services.

There are people locally who might be interested in speculative trading who would pay for the education -- not just market-timing trading on the exchanges but perhaps investing on MPEX, or ICBit.se (futures) as well.

There may be individuals who would love to play BitLotto, bitZino and SatoshiDICE but don't know what "provably fair" means.  So being able to demonstrate why these can be considered "provably fair" and explain how they can be used is a service that gamblers might pay for.

Bitcoin is just one of the components in the tool chest.  Just as knowing Bitcoin very well opens some doors, knowing other privacy-enhancing technologies can be valuable as well.   Tor and to a lesser extent mesh networking are two other skills a Bitcoin technology consultant will want to have expertise in.

There may be people willing to transact on the dark net but they don't already have the technical ability to do so.  Having someone local who is familiar with the steps (e.g., install Tor, install GPG, coin mixing, etc.) and can perform assistance in getting set up, in-person, is a valued service.  [Though be careful, and don't offer this service unless you really know your stuff.]  Consider joining or starting a CryptoParty:


http://www.CryptoParty.org


Local escrow is another service.  A person might not trust an online escrow but if they already know who you are, in real life, they might be more willing to use you as escrow for a trade.  

Somewhat related to escrow is to act as a buyer's agent.  For instance, someone is buying something from a seller in your town and wishes to ensure a good transaction.   So you accept the buyer's payment and then you meet the seller to inspect the goods before shipment. If the items are in working order you settle up (paying cash to the seller), and the shipment goes out.

Additionally, now that ASICs are just about to ship, a person can provide tech services to non-technical people.  The BFL equipment will still need a host, whether it is a Raspberry Pi or a desktop computer.  Unlike GPU and to a lesser extent FPGAs, the ASIC equipment does not require strong technical skills.  A miner can be set up in minutes, with only periodic maintenance.  Because there is not much heat produced nor any extraordinary connectivity requirements, ASIC mining is once again putting mining back in the range of capabilities of a typical slightly ambitious person.

Just look at all the ways bitcoins are used and there are unmet needs all over the place.