user81: How do I see the IP address that someone used to send or receive bitcoins with? Is it stored in the blockchain?

user40436: The IP information you can obtain monitoring the network traffic.  Block-chain doesn't contain any useful information to garner IP addresses, but if the user isn't careful with their activities or using a bitcoin mixer, tracking their block-chain info to ascertain habits won't be challenging, and you can use that to confirm an IP.
Now, onto bitcoin mixers.  If you use a clearnet mixer, those sites are subject to governmental regulation (which varies by government) and can be required to give up their records.  Darknet mixers on I2P and Tor are not subject to the same controls and don't typically record any information about their users.  Since there's no user data to surrender and the hash data from the block-chain is all scrambled, it's useless to LEOs.

user5: If the transaction was made through a third party website (such as en e-wallet or an exchange), you could try to contact the website owner and ask for the logs, if there are any.
The Bitcoin protocol does not record this information by itself.

user44: The blockchain doesn't store IP addresses.  In order to obtain the IP address of someone sending or receiving bitcoins, you would have either observe the activity of the network very carefully, or track them down by some other means.  Keep in mind that, for example, someone could receive bitcoins without ever being connected to the internet.

user99: No, IP addresses are not stored in the blockchain.
But Gavin Andreson indeed notes that

Unless you are very careful in the way you use Bitcoin (and you have
  the technical know-how to use it with other anonymizing technologies
  like Tor or i2p), you should assume that a persistent, motivated
  attacker will be able to associate your IP address with your bitcoin
  transactions.

One way to do this is to simply run bitcoin peer-to-peer clients that connect to a large fraction of the active nodes.  It's not as hard as you might think.  See Dan Kaminsky's Black Hat 2011 talk, starting at slide 21, for more.  (Thanks to kirian for reminding me of the link).  Then see An Analysis of Anonymity in the Bitcoin System 2011-07-22 by Fergal Reid, Martin Harrigan for how to link together the various bitcoin addresses that each user uses.
Some related background is outlined by Craig Reading:
Bitcoin Theft – The Top Ten Threats | Genesys Guru Blog

81: <p>How do I see the IP address that someone used to send or receive bitcoins with? Is it stored in the blockchain?</p>
 40436: <p>The IP information you can obtain monitoring the network traffic.  Block-chain doesn't contain any useful information to garner IP addresses, but if the user isn't careful with their activities or using a bitcoin mixer, tracking their block-chain info to ascertain habits won't be challenging, and you can use that to confirm an IP.</p>

<p>Now, onto bitcoin mixers.  If you use a clearnet mixer, those sites are subject to governmental regulation (which varies by government) and can be required to give up their records.  Darknet mixers on I2P and Tor are not subject to the same controls and don't typically record any information about their users.  Since there's no user data to surrender and the hash data from the block-chain is all scrambled, it's useless to <a href="https://criminaljusticeonlineblog.com/12/what-does-leo-mean/" rel="nofollow noreferrer">LEOs</a>.</p>
 5: <p>If the transaction was made through a third party website (such as en e-wallet or an exchange), you could try to contact the website owner and ask for the logs, if there are any.</p>

<p>The Bitcoin protocol does not record this information by itself.</p>
 44: <p><strong>The blockchain doesn't store IP addresses</strong>.  In order to obtain the IP address of someone sending or receiving bitcoins, you would have either observe the activity of the network very carefully, or track them down by some other means.  Keep in mind that, for example, someone could receive bitcoins <a href="https://bitcoin.stackexchange.com/questions/28/how-can-i-keep-my-wallet-secure">without ever being connected to the internet</a>.</p>
 99: <p>No, IP addresses are not stored in the blockchain.</p>

<p>But Gavin Andreson indeed notes that</p>

<blockquote>
  <p>Unless you are very careful in the way you use Bitcoin (and you have
  the technical know-how to use it with other anonymizing technologies
  like Tor or i2p), you should assume that a persistent, motivated
  attacker will be able to associate your IP address with your bitcoin
  transactions.</p>
</blockquote>

<p>One way to do this is to simply run bitcoin peer-to-peer clients that connect to a large fraction of the active nodes.  It's not as hard as you might think.  See <a href="http://www.slideshare.net/dakami/black-ops-of-tcpip-2011-black-hat-usa-2011">Dan Kaminsky's Black Hat 2011 talk</a>, starting at slide 21, for more.  (Thanks to kirian for reminding me of the link).  Then see <em>An Analysis of Anonymity in the Bitcoin System</em> 2011-07-22 by Fergal Reid, Martin Harrigan for how to link together the various bitcoin addresses that each user uses.</p>

<p>Some related background is outlined by Craig Reading:
<a href="http://genesysguru.com/blog/blog/2011/06/17/bitcoin-theft-the-top-ten-threats/">Bitcoin Theft – The Top Ten Threats | Genesys Guru Blog</a></p>
