请教我一个使用 来着 (different usage please) 的例句。 最好是在日常生活中经常使用的句子。

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I think it is very difficult for a foreign language learner to master the usage. Even in Chinese it is also very difficult to define clearly. Grammatically, it is a modal particle put at the end of a sentence signaling to an event that has happened or for an interrogative question.
他叫什么来着？ sorry, what is your name?
In this sentence the function of 来着means that you've told me once but I forget. Thus, I have to ask once again.
That way of speaking often signals a conversation context (来着 in most cases is used in conversations, I think) that the speaker has heard/mentioned/been told/known about the information in the coming conversation but forgotten or not been sure about it.
another example: 我说什么来着？sorry, what I was talking about? It may happen when the speaker forgot the point he was talking about during a conversation and had to asked the listener to continue his talk.

It may also be used in a statement
他去年冬天还回家来着 simililar to 他去年冬天还回家来了but it also refers to a certain context ahead of the this conversation. For example,上周他去世了，再也不能回去了。：He died last week and would never come back again.

It may also be used to ask pure questions.
你忘记小时候爸爸怎么教导咱们来着？Do you forget what my father had taught us? It may refer to a context that the listener didn't or didn't want to follow what his father taught him.

moreover, it may be used in rhetorical questions.

我说什么来着？他们比赛赢了吧？
In this they are both rhetorical questions. "They won the game as I had expected, didn't they？“ but it contains a strong underlying meaning that I was right.