My 5 year old female cat is very attention seeking. She will whine and whine (a really high sharp "mew" noise, almost sounds like she's saying "me me me me me me me me me") and she'll even push my legs to get attention.
Thing is when she is given the attention she'll run away and throw herself down on the floor out of reach.
If you just try to ignore her she just get more and more persistant in her "whining".

Should I continue trying to ignore her (she does get plenty of fuss when she isn't whining), or should I be pandering to her whims?

Hi does she do it all the time or occasionally? has she been spayed she could be in season if its on occasions.

Just seen your other thread,she has been done.one of my boys is very affecionate to the point of being in your face all the time,even up the table!

Yes, she's been spayed.
She does it daily, although not all day, more from mid afternoon, but she is doing it more and more as she gets older.

I wouldn't mind so much if she actually wanted fuss/attention/something but when you turn to her she runs away and throws herself down on the floor.

Furbabies eh?