to the best of one's ability是固定用法，不能用复数形式。类似的结构还有：

to my knowledge

to the best of my knowledge/belief

补充

我百度了一下，《有道词典》列举了很多to the best of my abilities的例句。看来确实这个短语的ability有用复数的。请选择刘老师的解答。

同意刘老师解答

A:to the best of one's ability（不可数名词，表示一种整体，抽象概念上能力）

I feel the personal satisfaction comes from doing work to the best of one's ability.  

我感到个人的满足感来自于尽力做好每项工作.

They tackled them to the best of their ability.  

他们竭尽全力给予解决。

B:to the best of one's abilities（可数名词，表示多方面，多种具体的能力）

I take care of them to the best of my abilities.  

我尽我所能地照顾他们。

Designed safety production administration system to the best of one's abilities conciseness understandability and practicality.  

设计的安全生产管理体系尽量做到简明易懂、实用.

to the best of one's ability=to the best of one's ability:尽…所能地；竭尽全力地

英文解释：If you do something to the best of your abilities or to the best of your ability, you do it as well as you can.

正如刘老师所分析的，这里的ability用单复数都对。请看Collins Dictionary提供的明确的解释：

If you do something to the best of your abilities or to the best of your ability, you do it as well as you can.

1. I take care of them to the best of my abilities.
2. They tackled them to the best of their ability.