I'm in the US and I would see it as a stripper name when used by caucasians and especially when spelled in an unusual way (Destinee, Destinye, Destynee, etc). When used by African Americans, it's something of a classic name. There are nuances in American naming trends that are not always apparent from across the pond - many names carry entirely different connotations when used in different settings within the US.
However, even given all of that, I don't think it's in the same category as names like Faith, Grace, Hope, or other virtues because it's not a virtue or an attribute or related to religion. It has an 'epic' quality to it as Florascotianew says, but that is a very worldly quality.
If you want a name along the lines of the three just mentioned, how about Joy? Or Serena? Or Letitia? (a name meaning joy/happiness)
Or Constance?
Vera means truth and also faith...
Honor and Justice are also nice names (imo).
Esme means 'loved' or 'held in high esteem'.
If you want a name with something of the same quality as Destiny, how about Stella, or Estella, or Estelle?
Or Esperanza, if Hope is already taken. Esperanza has a bit more oomph than Hope imo.
Similarly, if Constance as a virtue name appeals to you but you want to make it a bit more memorable, how about Constanza?
(I wouldn't use the name Serenity in the US - it's the name of an adult incontinence product).