Is the purpose of a woman's going to the mikvah so that she can permit herself to a man (be he her husband or any potential husband), or is it for her own sake? The nafqa mina is that of a woman who will never marry a man. Women who fall into this category today might include those who are in long-term lesbian relationships, but for a more classical example you can consider the woman who is twice (or thrice) widowed, according to Rebbi and Rabbi Shimon ben Gamliel, in Yevamot 64b. Can such women avoid going to the mikvah, or is it still a requirement?
Rabbi Isaac ben Sheshet was asked (Responsum 425) why no rabbinic edict requiring unmarried women to regularly purify themselves in the Mivka was ever enacted in order to minimize the transgressions of those who engage in extra-marital sexual contact. (I note the whole basis of the question is that in the days when women regularly or even semi-regularly were involved in handling sanctified food, such a purification was clearly performed. Nowadays that is not an issue [with the possible exception of women who would visit the outer parts of the Temple Mount].) He responds that no enactment was made because removing the biblical Karet-bearing prohibition of Niddah would allow people to grant themselves leniencies in this matter. The implication is that the longstanding custom was for women to not visit the Mikva outside the context of a "kosher" sexual relationship. Such is I believe the accepted practice today as well.
There is a Tannaic dispute whether or not going to mikveh at the earliest time possible is an independent mitzvah (טבילה בזמנה מצוה או לאו מצוה) (see e.g. Yoma 8a). According to this opinion, a woman would go to the mikveh irrespective of becoming permitted to her husband (as in Niddah 29b where a woman was sent to mikveh many times to account for many permutations of when her earliest time to go to mikveh was, all the while remaining forbidden to her husband).
The Beis Yosef (Y.D. 197 s.v. הלכך מצוה) cites several opinions that are explicit that according to this approach, a woman should go to mikveh even when her husband is out of town.
And, more directly to your question, the Shach (Y.D. 197:3) says that according to this position, unmarried women and widows should be going to the mikveh.
The halachic consensus seems to be that mikveh on time is not a mitzvah, as the Shulchan Aruch Y.D. 197:2 codifies that it is only a mitzvah when her husband is in town, as a function of the mitzvah of p'ru u'revu. However, several major Rishonim, including no less than Rabbeinu Chananel, held of this position.
It is wrong for unmarried women to go to mikva. I cant provide the source at the moment but will look for it. Since it makes them more 'eligible' for znus. Here פנויות לטהרתן, ומורים ליחידים ולעתים אף לרבים לילך ולטבול ומתירים את האיסור בק"ן טעמים", כתב הרב מצגר לרבנים. "אשר על כן בא אני בזאת לחזק ולבצר פסקם של ראשונים ואחרונים ולהודיע בשער בת רבים כי איסור גמור הוא לאישה פנויה, לטבול לטהרתה, וחובה למונעה ואסור לסייע בידה. והפורץ גדר ישכנו נחש". בעבר פרסמה הרבנות הראשית קריאות בנושא, וכמו כן פורסם מכתב של הרב עובדיה יוסף. קריאתו של הרב הראשי באה בעקבות שאלה שנשאל על רב שלכאורה התיר טבילת פנויה וכן בעקבות נוהג שהשתרש לאחרונה בקרב רווקות, בעקבות אמונה תפלה, ולפיו אם יטבלו במקווה וינהגו כמנהגן של נשים נשואות - יזורז זיווגן.
there is a custom among even single umarried men to go to the mikva every friday before shabbat.
I have heard people say they cannot feel the holiness of shabbat without going to the mikve.
I once read that Rabbi Chaim Kanievsky was asked if it is bitul torah (Wasted time) if a bachelor yeshiva student goes to the mikva frequently. he responded that on the contrary this is a good thing.
my point is tahara helps a person in his religious service. if this were not the case, yeshiva bachurim would not be encouraged to take time out from their learning. tahara helps a person connect to the spiritual (see shaarei kedusha gate 4)