I see over time there have been multiple suggestions of how to overcome the issue that b/i/u isn't always an option. The original question was regarding Notes, my issue involved Messages. I am certain what will work for Notes will also work for Messages, so this is relevant.
The thing is, once in maybe 20 or 30 selections (maybe even more) I see the B/U/I as an option. Unfortunately, those always come up when I have no need for them. However, when I need to bold/underline/italics, it's never there. And, I mean never with Messages and never with Notes. Formatting is simply not an option in any form. Also, when I turn my phone to the landscape position, at the left side there is a button for B/U, but it is always greyed out. I have never seen it available, so I wonder why it is even there.
And, I'm sorry, but I've tried all the "solutions" in the above replies, and none of them work for me in either Messages or Notes. I am on an iPhone 6+ with OS 8.4. I came here looking for the same solution as the original poster, and that was more than a year ago, but I hope Apple doesn't lock this conversation even though it is old, because it is a topic people are really curious about and based upon what is see thus far in the responses, this question clearly has no solution as yet.
I really need help here. Please someone, give us an answer! 😕