Is it considered an injustice for someone to sarcastically say behind my back that I have become an old maid, and is it permissible to supplicate against them?

If the description offends you and you dislike it, then it is backbiting. If it is accompanied by mockery and disparagement, the sin is greater, as it combines backbiting and mockery, as in His (Allah's) saying: "O you who have believed, let not a people ridicule [another] people; perhaps they may be better than them; nor let women ridicule [other] women; perhaps they may be better than them. And do not insult one another and do not call each other by [offensive] nicknames. Wretched is the name of disobedience after [one's] faith. And whoever does not repent - then it is those who are the wrongdoers." The Prophet (peace be upon him) defined backbiting as: "Your mentioning about your brother that which he dislikes." So, if what you say is true about him, you have backbitten him; and if it is not true, you have slandered him. Backbiting is a type of injustice against one's honor. The wronged person has the right to supplicate against their oppressor, but forgiveness is better. One must verify the news brought by a wicked person, in accordance with His (Allah's) saying: "O you who have believed, if there comes to you a disobedient one with information, investigate, lest you harm a people out of ignorance and then become regretful over what you have done." Marriage is among the good matters, so one should not despair if no suitor comes forward. One should supplicate frequently, as per His (Allah's) saying: "And when My servants ask you, [O Muhammad], concerning Me - indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided." It is permissible to search for spouses and offer oneself.