As of 2020

Go to this page, scroll down to the Other Notifications section, and make sure the option Replies to my comments is toggled on. There's no submit button - changes are saved the moment they are made, so simply toggling the option and closing the page will do.

I personally found that this setting was off on my profile even though I don't remember doing so, which makes me suspect that - for whatever reason - YouTube defaults to not notifying you of replies to your comments. Regardless, toggling this option on should mean you start receiving notifications for replies to your comments.

I know this is a fairly old posting, but there is a vastly different system in place now for YouTube comments altogether.

The above links will not work, and you will not be able to see anything there. To clarify a little further, the account settings only needs to be accessed if you are having trouble seeing your comments after you have sent or gotten a reply (on the video's page, not the account or history page.

Please go to this link if you would like to see your Commenting history - up to the entirety of the account.

https://www.youtube.com/feed/history/comment_history

You may also navigate about the area and locate several other very closely related topics, such as the "Watched Video History" at:

https://www.youtube.com/feed/history

Hopefully that can be of some assistance to someone in the future.

On YouTube, click your profile icon in the top-right and then click Inbox in the menu that appears. Messages and comment notifications will show up there.

You can also get to the inbox directly by going to http://www.youtube.com/inbox or go to the comment folder specifically with http://www.youtube.com/inbox#comments/

From the Change comment notification settings YouTube Help article:

You can manage whether or not you get email or mobile notifications about new comments and replies on your videos or channel.

Manage mobile notifications

Use your YouTube app to manage if you get notifications on your mobile device when someone comments or replies to a comment on your YouTube videos or channel. Learn more about managing mobile notifications for new content.

• Android
  1. In the upper right, tap the menu icon .
  2. Tap Settings > Notifications.
  3. Under "Other notification types," change your setting for Comments and replies.

• iOS

  Before you start: If you want to get mobile notifications, make sure that YouTube notifications are enabled for your device. Go to your device's Settings app > Notifications. Find YouTube in the list and make sure it's listed under "Include." If it's under "Do not include," tap it and turn on the switch next to Allow notifications.

  1. In the upper right, tap the menu icon .
  2. Tap Settings > Notifications.
  3. Under "Other notification types," change your setting for Comments and replies.

Tip: You can also do this with the Creator Studio app on your mobile device. Get started in the YouTube Creator Studio app Help Center.

Manage email notifications

Depending on your current account state, you may see one of two different ways to control the email notifications you receive when someone comments on your YouTube videos or channel or replies to one of your comments.

Channels associated with a Google+ profile or page

Use the Google+ profile or Google+ page connected to your YouTube channel to manage email notifications. You can also change what email addresses get notifications.

• Using a computer

  1. Go to www.youtube.com and sign in to your account.
  2. In the top right, click your account icon > settings icon .
  3. On the left, click Notifications to get to your account notifications.
  4. At the bottom, next to "Change email preferences for comments and replies," click Google+ notification settings.
  5. Under "Receive notifications," select Posts.
  6. Next to "Shares or comments on your content across the web," check or uncheck the box for Email.
  7. Under "Notification delivery," you'll see the email addresses that are currently set up to get notifications on your channel. You can add or remove email addresses here.

• Mobile site

  1. Go to m.youtube.com and sign in to your account.
  2. In the top right, tap the menu icon .
  3. Tap Settings > Email > Change on Google+.
  4. Under "Receive notifications," select Posts.
  5. Next to "Shares or comments on your content across the web," check or uncheck the box for Email.

Channels not associated with a Google+ profile or page

• Computer
  1. Go to www.youtube.com and sign in to your account.
  2. In the top right, click your account icon > settings icon .
  3. On the left, click Notifications to get to your account notifications.
  4. Select Send me emails about my YouTube activity.
• Mobile
  1. Open the YouTube mobile app.
  2. In the upper right, tap the menu icon .
  3. Tap Settings > Notifications.
  4. Tap Notify me via.
  5. Select an email option.

Tip: If you've previously opted out of email notifications, go to your account notifications settings and make sure that email notifications for comments isn't in the list of email types that you've unsubscribed from. If it is, just click the Resume link to start getting email notifications about comments again.

If you want to stop receiving email notifications about comments and replies, use the unsubscribe link in one of the emails to opt out of these types of notifications.