I needed to drill into a hard plastic box and didn't want to risk breaking it by using a center punch. I've had problems with the bit walking off the mark when starting on a hard or slick surface. This time I put some masking tape over the place I needed to drill. This gave the bit something to grip and helped to keep it on center.

RRM

Have you tried a center punch?



I also like using center drills in my drill press for metal or plastic chassis or boxes.



The center drill starts a very small hole without drift or wobble.

Rich

You posted just as I edited about using a center punch. I have the spring loaded kind. Sometimes they can't be used.

RRM

Start with the smallest, shortest drill bit that you own and chuck it way down in the chuck so that the tip cannot flex. If you can use a HOT sharp scribe to create a small divot for starting, that would also help.
EDIT : I see the suggestion for using a centre drill bit. That's about the best bit for starting on a slick surface.

All the follow up suggestions are good ones and will do a more professional job. A piece of masking tape works wonders and costs almost nothing.

RRM

Another option is an Exacto knife with common # 11 blade.

The spring load center punch usually has a chisel point and it is easy to punch the divot off by a 32nd of inch or so. Often that error is not important. With multiple punches in a surface that may add up. Project depending, I will start the divot with an Exacto knife then complete with the punch.

The masking tape idea does sound intriguing.

Brad point bits are real nice when you don't want the bit to walk.

For plastic I've heated the centerpunch and pushed in the plastic to form a depression, works pretty good.

Nick

My experience with plastic being too hard to center punch for fear of breakage is that it will also chip out badly with a standard twist drill, usually right at the point where it breaks through and grabs. A wood bit is actually better for plastic. I worked a job where they actually ground the twist drills ends to a brad point for that very reason to get odd sizes.

I've found the best way to drill plastic is as the OP did, using masking tape. I also use it on both sides, which prevents chipping as the bit goes through.

Allow the tip to show on the backside of the drilled hole, then drill from the backside with the brad point bit.

This is a carpenters trick when drilling with a worm auger: The preventing of the splintering of the hole on the backside. (Something my Dad taught me when I used his beautiful set of Russell Jennings augers.

My favorite solution is to use a fine point center drill, like a #1, in a drill press. Clamp the work piece and the hole should be perfect. I use the combination drills with a chamfer section, then use the correct dimensional bit.

Center drill have a thick shank to prevent wobble and drift.

Rich

When I cannot smack a punch with a hammer I've found these grinding attachments for my Dremel work perfectly.
My favorite is the fully globular one on the far left and it makes a perfect crater for drilling.

Doesn't seem very accurate if you have to control spacing between holes. I still like center-drills. The fine point is nearly a needle.

Rich

The original posting stated: "I needed to drill into a hard plastic box and didn't want to risk breaking it by using a center punch."

I just use a center drill for plastic at high speed and very slow feed.
Remember new plastic warm is less likely to break than old cold plastic. Here plastic I just drilled last month

You can use a screw machine drill {very short drill used in machine work} in a hand holder and start the hole

Dave

FYI
You drill plastic when was 10 it is call a soldering iron works great need to do outside in the wind.

Center drills are already short (2 to 3 inches) with thick bodies so they don't bend or wobble.

I also use screw machine drills, which are more stable, especially in small diameters.

Of course, I am using a drill press. I don't recommend hand drilling for any degree of precision.

Rich

The center drill does not fit the hand drill not electric.
You use drill instead of a center punch by drill a small spot in the plastic .

If have equipment you can make a holder for the center drill.

Dave