It depends on contexts, usually you would include honours status for your Bach., I use it on my CVs because it's an important distinction. I wouldn't use it after your name in letters, even professional ones, because it's pretentious. If you are going to include something after your name in a professional letter it should be something that is useful for locating you within a business, like your job title, instead of your credentials.
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Generally speaking, I think they're good selling points in resumes (unless, of course, the job might think you'd get "bored" by having the degree you do and so you would look for better offers in your spare time) and in an appropriate case it might show a relevant credential. But to just do it in ordinary emails and posts seems kinda like a lack of self-confidence to me and trying to prop yourself up (similar to someone who always goes on about how rich or strong he is or beautiful she is). I realize that's not necessarily the case (especially if you'd just earned it in which case it would look more like excitement and deserved pride rather than insecurity), but it would strike me as likely and not leave a good impression for me to start with.
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Laypersons wouldn't know what BSc(hons) AMSB meant. I wouldn't have known if you didn't say so. So, it'd only look good on a CV, really. Or on a business card. Or a name plate.
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