Zero in all steering parts. Some vertical in suspension but the noise with bother you. Rear suspension: no acceptable free-play. The only allowed deadspace is with tapered roller bearings because you want them to live. This all hangs on how fast you want to go. But, consider this: A tire that you might put on there may have a cornering stiffness of 800 to 1500 lbs/deg. That can really slap you around. And there are usually TWO tires on an axle, so go figure... Plus, because Quantum Mechanics creates entropy(shit flows downhill), All these parts will develop wear are tear, so its best to start out fresh & clean. All this does not mean "No Compliance". You need some to reduce shock loads and parts breakage (like your kidneys).

BTW: I won't even accept play in my lawnmower. I hate crooked cut lines.

A 2002 Miata with 205/45-R16 Bridgestones churned out about 250 - 300 lbs/deg. But you might be on race slicks that are FAR stronger.

My definition of the word 'play' in this context means free movement of one thing relative to another.

If we agree on that definition, the play in all bushings/ball joints etc needs to be zero. Any play will result in uncontrolled suspension movement, a lot of annoying noise, and highly accelerated wear.

What you need to be concerned with is compliance. Bushings act like springs (and often, dampers too) which add to the true wheel rate of whatever they are attaching to the chassis. The stiffer they are out of plane, the higher general harshness you will have, but the closer actual suspension movement will map to the expected result from a given spring/damper/geometry. And vice versa. The stiffer they are in plane, the more they will affect wheel rate. The best solution from first principles and disregarding NVH is a bushing that has very high out of plane stiffness, and almost zero in-plane stiffness; spherical bearings get relatively close.

As you add in out of plane compliance (ie, reduce stiffness) you get better NVH - ie better 'ride' and less harshness over bumps/impacts.

Ciba's comment regarding tire stiffness is valid, and will be a major factor in the end result. If it's truly a track-only build that's going to run slicks or high performance/low profile DOT tires, you will want to keep that in mind.

If you actually are modifying an existing chassis, your best result is going to come from buying off the shelf parts. Particularly for Miatas, there dozens of options in the range between street replacements and full bore hard core track parts. Trying to engineer and fabricate your own bushings is non-trivial engineering; the money and time you'd spend doing it yourself will be better spent elsewhere.

Fear not, but just because your Miata has about 50% weight distribution (which make for great advertising), it has gobs of front compliance steer, both lateral force and aligning moment. Front roll steer is 5%. The Mz steer slope (1.13 deg/100 Nm) is more than twice what a 'normal' everyday car would have. Same for the Fy steer. This is compliance, as in 'spring rate', NOT lash, slop, or worn parts. Not so great advertising, eh ?