Ideally, the list of your unread messages would be fully shown when you select the "INBOX (UNREAD)" view at the top left of the inbox popup/dropdown. However, the inbox popup/dropdown only shows the most recent 50 messages, so if you have a lot of messages, it's possible for some or all of your unread messages to not be displayed in the inbox popup/dropdown.
You can see all of your unread messages in your full inbox on stackexchange.com. There's a "Go to full inbox" link to your full inbox at the bottom of the inbox popup/dropdown. Within your full inbox, your unread messages will have a white background. There's no view of just the unread ones, so you will need to page through your inbox looking for the ones which are unread. Currently, there's no specific control within your full inbox to mark individual messages as read or unread. However, clicking on the link provided within a specific unread message will cause the message to be marked as read.
Once you've handled all of your unread messages, you can use the "Mark all as read" button in the inbox popup/dropdown to mark all your messages as read.
Note that, currently, just opening the inbox popup/dropdown on any of chat.stackoverflow.com, chat.stackexchange.com, or chat.meta.stackexchange.com will automatically mark all of your inbox messages as read.