Is a man allowed to listen to a woman on the radio? Would it change if you saw the actual girl (e.g. on TV singing)?
Yes you can listen to women on the radio but you cannot listen to a women sing. Kitzur shulchan aruch and possibly mishna Berura.
Kitzur Shulchan Aruch 152:8
The Sefer Avnei Yasfei 2:5:anaf 2 writes that its assur based off different sources he brings in the tshuva. He argues on those who allow it who he also brings in the tshuva. (it is worthwhile going through the whole Siman and all the anafim since he answers many kol isha questions which are commonly asked these days)
Rav Wosner in Shevet Halevi 3:181 also holds its assur.
Rav Yehuda Hertzel Henkin quotes his grandfather( Rav Yosef Eliyahu Henkin) in his Bnei Banim 2:Mamar 1:37 saying that there is no issur of kol isha on the radio,and he also one time asked regarding hearing a woman on tv and he said thats its not kol isha,but then he went back at a later time to ask again and he wasn't so clear if its ok(regarding the tv case).
Rav Yehuda Hertzel Henkin discusses in chelek 3:pg 115-116 ,pg 126 whether seeing a picture is considered knowing what the woman looks like or if one time is enough to be considered kol isha. However ,if one listens intently to hear the womans voice for pleasure then its assur according to everyone(see the tshuvas and sources, as well as the Mishna Brura siman 75:17).
For more sources see the sefer Piskei Tshuvos siman 75:12.
The Sefer Leket Hakemach Hachadash 75:3 brings a case about bentching infront of a tv. He sources this sefer Bais Dovid(Rav Leiter) siman 186 which brings both opinons of being machmir and being meikel.
Many poskim lean towards permiting it as long as you don't know what she looks like. See shu't Beis Shiarim siman 33, shu't Pri Hasadeh siman 32. Also see Chazon ish siman 15. These maareh mikomos are from She'arim Metzuyanim Bihalacha on maseches Taanis 5b. Also in his supercomentary to Kitzur shulchan Aruch siman 5 s.k. 14. He adds however that having seen a picture counts as knowing what she looks like. There is also a tshuva in Yabia omer permitting this. I don't have maareh makom though.
Rav Ovadia Yosef writes (Yabia Omer 9:108:43) that a woman's voice on the radio is not Kol Isha. He cites in agreement Rs Yosef Shalom Elyashiv and Yitzchok Zev Soloveitchik.