Couple of options

You could screw it to the top of the dresser and hide the screw heads under removable scenery/buildings. Unscrewing it to move it would be pretty easy. You only really need two or three screws if you are just worried about bumping it.

If the layout is larger than the top of the dresser you could attach cleats to the bottom of the layout that would fit over the edges of the dresser top.

Bottom boards

Hi Kaitlin,

One option - you could attach small strips of wood to the bottom of the layout, slightly larger than the top of the dresser, and extending down a couple of inches.  These would server to keep the layout from moving due to an occasional bump, but you could still lift the layout off the dresser.  I'd add something soft between the layout and dresser too, like felt, to prevent scratching the dresser finish.

Rug Underlay

Reading twforeman's reply I realized the layout might be smaller than the top.  Another idea would be to put rug anti-slip pad under the layout.  This would make it slightly "sticky" and keep it from moving from occasional bumps, and from moving too far if pushed.

https://www.amazon.com/Rose-Home-Fashion-Non-Slip-Rubberized/dp/B06Y43HDVV/ref=sxin_3_ac_d_rm?ac_md=1-1-Y2FycGV0IGdyaXBwZXIgZm9yIGFyZWEgcnVncw%3D%3D-ac_d_rm&crid=2LIX5M993XKUU&cv_ct_cx=rug+anti+slip+pad&keywords=rug+anti+slip+pad&pd_rd_i=B06Y43HDVV&pd_rd_r=f483353b-06a7-4f3f-a7a1-39a5ff2b0b07&pd_rd_w=elJtV&pd_rd_wg=4yvrD&pf_rd_p=6d29ef56-fc35-411a-8a8e-7114f01518f7&pf_rd_r=21JQY60JHG2BAQCHAJ82&psc=1&qid=1579879011&sprefix=rug+anti+%2Caps%2C143&sr=1-2-12d4272d-8adb-4121-8624-135149aa9081

Velcro?

Velcro?

any idea what kinda screws i

any idea what kinda screws i would need?

Kaitlin, any kind that is long enough to go through the MDF and

Stick out about 1/2” to 3/4”. 1”-1-1/2” drywall screws would work. 

3M Command Strips

Jim's Velcro suggestion got me thinking 3M Command Strips with the 8 pound rating might work.  They are removable so you don't need to drill or worry about damaging the dresser. Get the refill pack as you don't need the hooks. Place the longer strip (wall side) on the dresser top (that strip are easier to completely remove if you need to) and the other strip on the bottom of the layout.  You'll probably need to measure and mark the layout bottom to find the right location for the strips.  I think one set near each corner of the layout should be sufficient to hold a N scale layout in place.  Push the layout down tight and you're good to go.  If it doesn't work, you're only out about  $10 USD.

Lou in California.

Dresser top?

If you don't want to damage the dresser top you could get metal T or L brackets and attach them to the back of the layout board and down to the back of the dresser. No need to screw anything into the dresser top as the rear attachments would keep the layout from sliding around. An alternative would be double sided foam tap but I'd check a small bit on the dresser top first to make sure it would peel off without damaging the finish......DaveB