They all mean "it seems/appears that ...", but each has better suited usage cases. ~것 같다 (it seems) is the most general. It can be used in diverse situations including both direct perception-based reactions and deduced conclusions. But because of this generality, it's sometimes not as effective or to the point as the other phrases. ~아/어 보인다 (it appears) when you say what you notice from seeing something, either directly or by making a further guess. It usually does not involve much thought process. ~ 모양이다 (I reckon/guess) indicates what one surmises from the information one has. It has the sense of "Judging from the way things are, it looks like ...". It feels like a statement of a guess and judgment compared with the other phrases. Most usages of 보인다 and 모양이다 are restricted to the third person only because people generally don't guess or deduce something about themselves or the listener. There is also ~나 보다 and ~(는)가 보다 (I guess/notice) which has elements of both ~아/어 보인다 and ~ 모양이다. Examples: * 날씨가 추워지는 것 같아 / ... 추워지는 것처럼 보여 (slightly unnatural) / 추워지는 모양이야 / 추워지나 봐. * 좋은 시절은 다 지나간 것 같다 / 지나갔나 보다 / 지나간 모양이다 = It seems the good times are all past. * 나 쓰러질 것 같아 = It looks like I'm going to collapse. (It's first person, so 보여, 모양이야 won't work). * (너) 안색이 안 좋아 보여 / 안 좋은 것 같아 = Your complexion looks bad. * (너) 무슨 근심이 있어 보여 / 있는 것처럼 보여 / 있는 것 같아 = You appear to be worried about something. * 나는 근심이 있어 보여 = I appear to be worried (UNNATURAL - because we don't guess about ourselves). * 나 너무 뚱뚱해 보인다. = I look too fat (Okay - because it's about physical appearance not involving guessing). * 아빠가 새 사업을 시작하실 모양이야 / 시작하시려나 봐 = It looks like dad's going to start a new business. * 이 동네에 고층 건물이 들어설 모양이야 / 들어서려나 봐 = It appears a high rise building will go up here. * 쟤네 둘이 싸우려나 봐 / 싸우려는 것 같아 = Those two boys seem to be going at each other.

It's also important to note their correct forms as used in real life, because they are not all the same in construction. * 같다 is an adjective, so 같습니다, 같다, 같아(요) (같는다 doesn't exist). * 보이다 is a verb (in passive voice), so 보입니다, 보인다, 보여(요) are possible. '보이다' is the dictionary form which (almost) never appears in real usage. * 모양이다 is made of 모양(n) + 이다(special particle). The whole phrase acts like an adjective. * 보다 in '~나 보다' is an (auxiliary) adjective, so only 봅니다, 보다, 봐(요) are possible. The verb conjugation form 본다 is not possible. For example, it's 비가 오나 보다, not 비가 오나 본다. The latter has an entirely different meaning.

Korean use those same situation because all 3 have similar meaning and can be translated as "seem" and hardly have different nuance. *~것 같다 be likely to, feel like~, i think that ex) 비가 올 것 같다 : it's likely to rain i think that it will rain * ~처럼 보이다 literal meaning is "be seen like~" real meaning is "look (like) ~" ex) 비가 올 것처럼 보이다 it looks like raining *~ 모양이다 literal meaning is " it's shape(or situation)of ~ real meaning is "seem" (i'm sorry i want to use another word but i can't find similar usage in English except "seem") ex) 비가 올 모양이다 it seem to rain (judging from shape of cloud)