Difference between IPA ɚ, ɹ, and ɝ

An /ɝ/ is just the stressed version of an /ɚ/. For example, murder has both of them in it, being normally written as /ˈmɝdɚ/. Both of those are “r-colored” vowels. However, some transcribers prefer to represent that as /ˈmɜɹdəɹ/ instead, writing a consonant instead of little rhotic hook. Those represent the same pronunciation.

Your mother is therefore going to be either your /ˈmʌðɚ/ or your /ˈmʌðəɹ/. You need to understand though that /ɚ/ and /əɹ/ are just two ways of writing the same thing — at least in words like murder and mother. When you can get into words like murdering or mothering, then you cannot use the r-colored version for the one before the -ing, since it now has a vowel after it and so much be written as a consonant.

There are advantages and disadvantages to doing it one way or the other. Using a consonant instead of a diacritic can be easier to understand, since you don’t have to think about whether it has a consonant following it and so counts as a rhotacized vowel, or whether it has a vowel following it and so counts as a consonant.

• mirth:          /ˈmɝθ/      or  /ˈmɜɹθ/
• mother:       /ˈmʌðɚ/      or  /ˈmʌðəɹ/
• mothering: /ˈmʌðəɹɪŋ/     
• murder:       /ˈmɝdɚ/      or  /ˈmɜɹdəɹ/
• murdered:   /ˈmɝdɚd/    or  /ˈmɜɹdəɹd/
• murderous: /ˈmɝdəɹəs/ or  /ˈmɜɹdəɹəs/
• murderer:    /ˈmɝdɚɚ/    or  /ˈmɜɹdəɹəɹ/

Another issue is that IPA doesn’t have special precomposed characters for other rhotacized vowels, so you have to build the others yourself, which means they don’t look like the precomposed ones:

• Mordor:               /ˈmo˞do˞/ or  /ˈmoɹdoɹ/
• corner:                 /ˈko˞nɚ/   or  /ˈkoɹnəɹ/
• harder:                /ˈhɑ˞dɚ/    or  /ˈhɑɹdəɹ/
• radar:                 /ˈɹeɪdɑ˞/   or  /ˈɹeɪdɑɹ/
• carport:              /ˈkɑ˞po˞t/ or  /ˈkɑɹpoɹt/
• rarer:                  /ˈɹeɪɹɚ/    or  /ˈɹeɪɹəɹ/
• creature:             /ˈkɹitʃɚ/  or  /ˈkɹitʃəɹ/
• entrepreneur:     /ˌɔntɹəpɹəˈnɚ/ or /ˌɔntɹəpɹəˈnɜɹ/
• entrepreneurial: /ˌɔntɹəpɹəˈnʊɹiəl/

If you are doing phonemic transcriptions, you might consider just sticking with /r/ and not worrying about all the various phonetic realizations possible for it, including [ɝ], [ɚ], [ɹ], [ɻʷ], and all the rest.

As for trying to identify difference between the several Canadian accents and the many American ones, when it comes to your r’s, this mostly depends on whether you are comparing rhotic dialects with non-rhotic ones. Note also that the standard versions of both sets are rhotic. That means your mother is still going to be the same wherever you are, and that you don’t need to worry about it. The mother of Vancouver is the same as the one from San José.

You haven’t said whether your first language is some variety of English, or whether it is something else. If it is something else, especially one without the sorts of rhotics that occur in North America, then simply mastering those alone will be much harder, and much more important, than trying to tease out one or another difference between this or that American or Canadian accent.