Everything I've read has said that there is no universal static reference frame, but based on my understanding of special relativity I don't understand why it can't be determined.
Here is an experiment I can think of that would seem to determine "universal static", what am I missing? Assuming 1 dimensional space for a second, our reference frame (a spaceship) is moving and an indeterminate speed $V_0$. We take two probes with atomic clocks and send one in each direction at some fraction of the speed of light ($V_0+V_p$ and $V_0-V_p$). After a period of time, the probes return at a slower speed than they left (taking advantage of the fact that time dilation effects increase exponentially with velocity). Upon returning, the clock in the probe sent in the positive direction ($V_0+V_p$) should show less time past than the clock on our spaceship, and the probe sent in the opposite direction ($V_0-V_p$) should show that more time has past than the clock on our spaceship, this tells us which direction we are moving with respect to "universal static", and with repeated experiments we can determine our exact velocity relative to it.
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