As to the question of when to use はほしい　[when it is ほしい　and not ほしくない]:

I think that @l'électeur gave one good example (=when emphasizing the contrast of what one wants and what not) with the example 「靴は欲しくないけど、ジーンズは欲しい。」

IF YOU READ FURTHER, PLEASE READ ALSO COMMENTS GIVEN AS IT SEEMS BELOW MAY BE INCORRECT. I WILL EDIT ONCE I FIND OUT MORE!

In addition I can think of at least two other situations to use は but at least I clicked on this question to find out more, so I hope the natives / gurus [also others than @l'électeur ] correct / add:

Can be used like in a [phrase or at least close] "あれはほしいこれはほしい" as a critical statement about someone with too many requests [and here it is more about requests than aspirations], i.e. in a stereotypical case two employees could talk about their boss [or similarly 2 managers could talk about someone's difficult team member] who is never satisfied and comes up with continues and unreasonable requests.

Can be used when the target of the aspirations is a bit abstract and/or not immediately achievable / obtainable, like I want that kind of life [ああ言う人生はほしい]

As to the question of when to use はほしい　[はほしい　and not ほしくない]:

I think that @l'électeur gave one good example (=when emphasizing the contrast of what one wants and what not) with the example 「靴は欲しくないけど、ジーンズは欲しい。」

IF YOU READ FURTHER [NOT ADVISABLE, BUT I DON'T DELETE THIS

1. Due to the valuable comments

2. To indicate when は欲しい IS NOT USED ;-)

3. To call for additional case where は欲しい　(and not e.g. が欲しい or は欲しくない)　is used

]

In addition I can think of at least two other situations to use は but at least I clicked on this question to find out more, so I hope the natives / gurus [also others than @l'électeur ] correct / add:

Can be used like in a [phrase or at least close] "あれはほしいこれはほしい" as a critical statement about someone with too many requests [and here it is more about requests than aspirations], i.e. in a stereotypical case two employees could talk about their boss [or similarly 2 managers could talk about someone's difficult team member] who is never satisfied and comes up with continues and unreasonable requests.

Can be used when the target of the aspirations is a bit abstract and/or not immediately achievable / obtainable, like I want that kind of life [ああ言う人生はほしい]

As to the question of when to use はほしい　[when it is ほしい　and not ほしくない]:

I think that @l'électeur gave one good example (=when emphasizing the contrast of what one wants and what not) with the example 「靴は欲しくないけど、ジーンズは欲しい。」

In addition I can think of at least two other situations to use は but at least I clicked on this question to find out more, so I hope the natives / gurus [also others than @l'électeur ] correct / add:

Can be used like in a [phrase or at least close] "あれはほしいこれはほしい" as a critical statement about someone with too many requests [and here it is more about requests than aspirations], i.e. in a stereotypical case two employees could talk about their boss [or similarly 2 managers could talk about someone's difficult team member] who is never satisfied and comes up with continues and unreasonable requests.

Can be used when the target of the aspirations is a bit abstract and/or not immediately achievable / obtainable, like I want that kind of life [ああ言う人生はほしい]

As to the question of when to use はほしい　[when it is ほしい　and not ほしくない]:

I think that @l'électeur gave one good example (=when emphasizing the contrast of what one wants and what not) with the example 「靴は欲しくないけど、ジーンズは欲しい。」

IF YOU READ FURTHER, PLEASE READ ALSO COMMENTS GIVEN AS IT SEEMS BELOW MAY BE INCORRECT. I WILL EDIT ONCE I FIND OUT MORE!

In addition I can think of at least two other situations to use は but at least I clicked on this question to find out more, so I hope the natives / gurus [also others than @l'électeur ] correct / add:

Can be used like in a [phrase or at least close] "あれはほしいこれはほしい" as a critical statement about someone with too many requests [and here it is more about requests than aspirations], i.e. in a stereotypical case two employees could talk about their boss [or similarly 2 managers could talk about someone's difficult team member] who is never satisfied and comes up with continues and unreasonable requests.

Can be used when the target of the aspirations is a bit abstract and/or not immediately achievable / obtainable, like I want that kind of life [ああ言う人生はほしい]