In 1999, were most people listening to cassettes or CD’s?

CDs. They first came out in the mid/late 80s, and by the early 90s had pretty much become the most popular medium for music. Cassettes were of course still being made, but CDs overtook them pretty quickly and remained the format of choice until MP3s came in.

I started listening to CD's in 1995.

CD's. It was earlier in the decade when the popularity of CD's surpassed cassettes. In the 80's, a lot of people couldn't afford CD's.

CD's, they were selling more CD's than cassettes by the mid 1990's 

Cassettes were really never the main way of listening to music at any stage. They never actually outsold vinyl. They existed in tandem with vinyl for about 20 years, but by 1999, they were already pretty old fashioned.

By then, cassettes had had their time. They were still on the shelves in the shops but when you looked at them you really thought to yourself who the heck is going to buy those?

CDs. CDS pretty much dominated in the 90s. Although people still had their cassette and vinyl collections. Around about 99 dvds were beginning to overtake VHS as well. 

Most of the music I listened to back then was on a CD, but I had a pretty extensive vinyl collection and would listen to records pretty often. I still had a good number of cassettes - mostly stuff that I couldn't find on CD, but most cars weren't equipped with a CD player until later, so when I was in the car it was always cassettes. Eventually I was able to get my hands on a device that enabled me to play CDs through the tape deck, which was a real game-changer. 

CDs had killed vinyl by then, but there was no replacement for cassettes.  DAT was too expensive for most people. Recording engineers and well heeled audiophiles adopted it but most consumers did not. Cassettes finally died when CD-R and CD-RW disc and CD recordable drives were invented in the late 1990s. MP3 also became available by then, and people can jog while listening to music with an MP3 player instead of a protable cassette tape player. 

There's never been a time really when, “most people” were listening to cassettes. First there were records and then, introduced in 1984, Compact Discs. After maybe ten years or so CDs had more or less taken over from records.

Cassettes existed in parallel and had two advantages: they could be recorded on - many people borrowed albums (or later CDs) and made their own cassette copy, and, of course, you could record from the radio. Cassettes were also portable - they could be played in a car and portable cassette players were available (Walkman).

Obviously there were exceptions, but people who wanted to listen to reasonable quality music used a record deck or, later, a CD player. Many people had both and most had cassette recorders too.