As part of a large-scale search for coherent pulsations from LMXBs in the RXTE archive, we have completed a detailed series of searches for coherent pulsations of 4U 1820-30 -- an ultracompact LMXB with a binary period of 11.4 min, located in the globular cluster NGC6624. The short binary period implies any coherent signal would be highly accelerated, so we used phase modulation searches, orbital-parameter-fitting coherent searches, and standard acceleration searches to give significant sensitivity to millisecond pulsations. We searched, in four energy bands and at a range of luminosities, a total of 34 archival RXTE observations, 32 of which had on-source integration times longer than 10 ks, and some of which were made consecutively which allowed us to combine them. We found no pulsations. Using our phase modulation search technique, which we ran on all 34 observations, we have been able to place the first stringent (95% confidence) pulsed fraction limits of <~0.8% for all realistic spin frequencies (i.e. <~2kHz) and likely companion masses (0.02Msun <= M_c <= 0.3Msun). Using our orbital-parameter-fitting coherent search, which we ran on only 11 selected observations, we have placed a pulsed fraction limit of <~0.3% for spin frequencies <~1.25kHz and companion masses M_ <= 0.106Msun. By contrast, all five LMXBs known to emit coherent pulsations have intrinsic pulsed fractions in the range 3% to 7% when pulsations are observed. Hence, our searches rule out pulsations with significantly lower pulsed fractions than those already observed.
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