When I started singing I used to sing in the lower register of my voice as it felt comfortable and is most like my speaking voice - so it's what I'm used to hear. If I sang higher then my voice started sound funny and light to me so I used not to do it. Record your voice and see how it actually sounds - don't worry about pitch or anything just what the notes sound like and you might find that you produce a much nicer sound higher than YOU think you do. (My speaking voice is reasonably low - if you listen here the first voice is mine)
A bass voice goes somewhere from a D up to about middle C - which was about where I would sing. As I've got used to my voice and practiced a bit and used it, the range has shifted upwards and I find I sing things now which take in much higher notes (up to about G above middle C) than I used to and that I actually prefer the tone of my voice in the higher range rather than when I was dropping an octave and mumbling away.
I try and sing All Right Now in A and that I do find a struggle given my voice and the range of the song - lowest note is an F# and the highest a C natural one and a half octaves above - warmed up I can probably hit the A but not the C. I have tried singing it an octave lower which easily fits in my range but it doesn't sound right. So I usually just change the top bit a little so that I can do it.
If you want to be Frank Sinatra then singing in predominantly a chest voice is great. For pop or rock etc it's worth experimenting as I think they normally refer to the range from about D through to C at the top end as the money range.
A lot of it is down to how you hear your own voice - record it and you may find you want to sing elsewhere.
Once my voice broke, Hark the Herald Angels at Xmas used to be the one where I wasn't sure what to do. Normally it is done in G and the highest note of the tune is an E - and I was never sure whether I could get there or not so ither throught or at a certain bit I would drop an octave which doesn't really work! The E is easy to sing now so it wouldn't be a problem if I wanted to sing Xmas carols.
Practice and experiment. If you can sing in tune and are happy in one part of your voice just practice and try other bits of your voice.
Lee Marvin on Wanderin Star worked well