From what I've read, Koreans use the word '접속사'(接續詞 - Conjunction) when discussing English conjunctions. To be really exact, the examples you gave have more complicated labels than just '접속사'. For the following, I consider:
- -아서/어서 to be 연결어미(連結語尾 - 'Connecting Ending').1
- -니까 to be 어미(語尾 - 'Ending').
- -때문에 - Not precisely sure. I know that 때문 is 의존명사(依存名詞 - 'Bound Noun'), and that -에 suffix is 조사(助詞 - Particle).1 Not sure what I would call them added together.
- -는 바람에 - -는 to be 어미. 바람 is 의존명사, and so on.
- -느라 to be 어미.
'어미' commonly means mother, but not here; the Hanja differs. It is a label used for ending words; their usage depends upon the application of predicative and predicate words within the sentence.
I nitpick your examples in order to show their nuanced difference, as you said in your title, and to show the degree of difficulty in your request to know which ones are used more commonly compared to others. One can also clearly see the limits of English grammar labels when used on deeply foreign languages.
Oh, and just to add: Examples that I consider to be '접속사' include 그러나('However') and 그런데('By the way'). Strictly put, they are not labeled as '접속사' but '접속 부사'(接續副詞 - 'Conjunctive Adverbs'). I make this distinction in order to clearly answer your question; I myself would not bother with it much. At any rate, the 표준국어대사전 considers '접속사' and '접속 부사' to be one and the same.
So what are the most commonly used 어미's? Can't say for sure, unless I research for several more days. But I can outline the general types. Now, this is going to be long; please bear with me here.
There are two types of 어미: 곡용어미(曲用語尾 - 'Declension Ending') and 활용어미(活用語尾 - 'Conjugation Ending').1
곡용어미 attaches to 체언(體言 - 'Uninflected Word'/'Substantive'), another word for 곡용어간(曲用語幹 - 'Declension Stem').1 활용어미 attaches to 활용어간( 活用語幹 - 'Conjugation Stem'1/'Verb Stem').
곡용어미's that are attached to 체언's are mainly 격어미(格語尾 - 'Case Ending').1 There are many different types of 격어미:
- 주격(主格) - Nominative Case
- 관형격(冠形格) - Genitive Case
- 목적격(目的格) - Objective Case
- 부사격(副詞格) - Adverbial Case1
- 열거격(列擧格)/접속격(接續格) - Conjunctive Case(Not sure at all about this one. 접속 means 'access'. 열거 means 'enumeration'.)
- 호격(呼格) - Vocative Case
Here, 부사격 is further divided up into these types:
- 여격(與格) - Dative Case
- 처소격(處所格) - Locative Case
- 탈격(脫格) - Ablative Case
- 조격(造格) - Instrumental Case
- 비교격(比較格) - Comparative Case
(I haven't really appreciated the diversity of Korean language up till this point.)
In special cases, some types of 격어미 are relabeled as 격조사(格助詞 - 'Postpositions').1
활용어미's that are attached to 용언어간(用言語幹 - 'Predicate Stem')1 are categorized into following types:
- 종지형(終止形) - Termination Type1
- 접속형(接續形) - Connection Type1
- 자격형(資格形) - Qualification Type1
종지형 has several 어미 types:
- 평서법(平敍法) - Impartial Mood1
- 의문법(疑問法) - Inquisitive Mood1
- 감탄법(感歎法) - Exclamatory Mood1
- 명령법(命令法) - Imperative Mood
- 청유법(請誘法) - Suggestive Mood1
- 응낙법(應諾法) - Compliance Mood
접속형 is divided into two categories:
- 등위접속(等位接續) - Coordinate Conjunction
- 종위접속(從位接續) - Subordinate Conjunction
자격형 has several 어미 types:
- 부사형어미(副詞形語尾) - Adverb Form Ending1
- 관형사형어미(冠形詞形語尾) - Modifier/Adnominal Form Ending
- 명사형어미(名詞形語尾) - Nominalizer Ending
With 어미 there are also 어말어미(語末語尾 - Final Ending) and 선어말어미(先語末語尾 - Prefinal Ending); I could go more into them if you wish.
So far I've only listed general categories without going into too much detail, as this answer is long enough as it is.
I now realize that the question itself is broad enough for me to possibly write this long without actually answering your essential question. If so, please inform me so I can add more to this answer.
1: This is my own translation. Take it with a grain of salt.