I recently acquired an old 96 Stumpjumper and after riding it for awhile I noticed it has a bit of play when I apply back and forth pressure to the handlebar which doesn't seem right to me. Am I correct in assuming it might be a problem with the headset? This is the first bike I've owned with a threadless headset so I don't know much about them. I've been looking at some photos of different headsets on google images and they all appear to be different than mine. Is it just me or shouldn't it have a top cup covering the black rubber washer?

Without knowing the brand of the headset, it's hard to tell if it's missing parts. If there are no exposed bearing, it's probably not as in your case.

For the play you are experiencing, all you should need to do is tighten up the headset, details on how to adjust can be found on the Park Tools website https://www.parktool.com/blog/repair-...service#adjust

Thanks, I was planning to try that but wanted some opinions on the appearance of the headset before I do. I just looked it up and according to bikepedias page for the bike its a Tange-Seiki headset.

I looks complete to me also. BTW, be sure the upper edge of the stem is 2-3 mm above the top of the steerer tube. If they are flush, you won't be able to properly preload the headset and will need to add a spacer either below or above the stem to obtain the adjustment gap.

What does the bottom half look like? Almost looks like the stack there is both of the cups that are pressed into the frame mated to each other.

It does look like that doesn't it? I think the upper cylinder is just a 10 mm or so spacer.

I think so too- it does look like the bearing cup, meant to be pressed into the headtube. If that's the case there is no way to "lock" the bearing to the steerer. Me thinks someone got their assembly procedure bass-ackwards.