For translators all of those verbs are the verb to use in English. Is it correct?
"Использовать" is the most universal word for "to use", and also for "to exploit" and "to utilize". It can be used for any kind of inanimate things, both material items and non-material things (like chances, capacities, technologies and so on). For example:
В нашей работе мы используем новейшие технологии, современное оборудование и высококачественные материалы.
In our work we use the newest technologies, modern equipment and high-quality materials.
But when "использовать" is used for people, it means that one treats the person like an inanimate thing, not a human. Of course, it sounds negative. For example:
Я считал его другом, но он просто использовал меня для своей выгоды!
I claimed that he was my friend, but he just exploited me to his own benefit!
"Поиспользовать" sounds awkward and is used rarely. But when it is, as @Justin McGuire already noticed in his answer, it usually means the same as "использовать" about people. For inanimate things it is almost never used, just use "использовать" instead.
"Употреблять/употребить" have more narrow meaning than "использовать". It is not used for people, for non-material things (except for words and grammatical constructs) and for material things, which remain intact after usage, and can be used again later. The meaning of "употреблять" is closer to "to consume". It can be used for food, drinks, drugs and other things we take inside our body. It can also be used for different kinds of supplies, which transform or disappear after usage, for example:
Эти доски давно без дела лежат, занимают много места. Надо бы их на что-то употребить...
These planks have been lying here for long time, occupying much space. They should be used this or that way...
As mentioned in @Justin McGuire's answer, "употреблять/употребить" is also used for words and grammar categories. Examples:
1) Это матерное слово, не употребляй его в приличном обществе!
This is a swear word, don't use it in polite company!
2) Какой падеж здесь употреблять - родительный или винительный?
Which case should be used here - Genitive or Accusative?
"Использовать" can be used instead of "употреблять" in all these cases, except for things you take inside and for words. Grammatical categories and supplies allow both verbs.
If you have doubts, which verb to use, it's better to use "использовать", because it is correct in most cases.
"употреблять/употребить" is about a grammatical aspect. "Употребить" is a perfect aspect (meaning that it was a single action). E.g. Yesterday I used (употребил) that f-word (I mean I did it once). "Употреблять" is an imperfect . E.g. All my childhood I used (употреблял) only good words (meaning it was more than once, lots of single actions for some period of time).
"Использовать" a bit differs from "употреблять". "Употреблять" is usually used with "words" (meaning saying words) and with something that goes inside you (e.g. about food - употреблять в пищу). "Использовать" (btw, omonynic forms - both imperfect and perfect aspects depending on the context) is just "use".
And "поиспользовать" sounds a bit negative (very rarely about inanimate, usually about people), e.g. "Поиспользовал и бросил" - about a girl who was later dumped by a douche.
These words are often interchangeable, but there are some established expressions in which one word is preferable to the other. They are also similar etymologically (cognate with потребность и польза, respectively).
Поиспользовать from the title of your question is redundant and awkward with both prefixes по- and ис-. The suffix по- instead of ис- here would mean the speaker is an oaf who had his way with someone. That's just the only context that comes to mind. Although one could use the other words in that context.
Употреблять somewhat implies consuming (e.g. food), and использовать, пользоваться implies using something without consuming it (e.g. a plate) . Also, использовать sounds more modern and is used (no pun intended) when talking about science and technology, for example.
Another difference is that there is a cognate reflexive verb пользоваться ("я пользуюсь зубной щеткой"), but no similar form with употреблять; the verb употребляться is only used in passive voice ("зубные щетки не употребляются в пищу").
Some established expressions with the word употреблять, as Justin McGuire pointed out, are употреблять в пищу, употреблять выражения.
A bad translation from English will usually be использовать.