请问关于英语翻译和语法的问题。

你应该查rest on的意思
断句是这样的
1.Billie Holiday’s reputation
2.as a great jazz-blues singer
3.rests on her ability to give emotional depth to her songs.

rest on
1.搁在〔支撑在〕…上
2.依赖于, 寄托于…上, 基于
3.接触到; 达到; 朝向; (使)落在…上 1.搁在〔支撑在〕…上
The bridge rests on stone arches.
这座桥由石拱支撑。
Her head rested on one of her hands as she sat there waiting.
她一只手托着头, 坐在那儿等候。
2.依赖于, 寄托于…上, 基于
Your success rests on your efforts.
你的成功有赖于你的努力。
His argument does not rest on reasoning or on experiment, but on authority.
他的论点不是依靠推理或者实验, 而是靠权威。
We rest our hopes on you.
我们把希望寄托在你们身上。
3.接触到; 达到; 朝向; (使)落在…上
His eyes rested on the peaceful valley below.
他的目光落在下面宁静的山谷中。
It's good to rest one's eyes on such a beautiful view.
观看这样的美景使人赏心悦目。

Billie Holiday’s reputation as a great jazz-blues singer rests on her ability to give emotional depth to her songs.

singer rests不是一个词语
singer单独翻译为歌手 rest on 才是一个词组 意思为“依靠、依赖于”

随意应该在singer那里断句才对。

这句话中。
Billie Holiday’s reputation as a great jazz-blues singer是整句话的主语。
as a great jazz-blues singer 是reputation的修饰语。补充说明了这个人在哪一方面的声望。
rests on 是句子中充当谓语动词的成分。
her ability to give emotional depth to her songs. 充当宾语。
the ability to do sth是固定句型。译为“做某事的能力”
to do是不定式的形式。

这句的主语是Billie Holiday’s reputation , as a great jazz-blues singer 在此做了singer的定语限定了它，rest 指“位于。。。或在于。。。”，你的翻译里它翻译成了“建立在。。。”

Billie Holiday’s reputation as a great jazz-blues singer rests on her ability to give emotional depth to her songs.

rests on 这应该是一个短语，此处翻译为建立在。。。