How significant is RNA degradation with removal of cap/polyA's in eukaryotes, or UTR's in prokaryotes?
Removal of 5' cap is essential for degradation by 5'→3' exonucleases such as Xrn1/2. Xrn1/2 is constitutive and degradation of uncapped RNAs would be quite fast (don't have a reference for the exact lifetime). Deadenylation generally precedes 3'→5' degradation by exosome but I am not sure if that is a prerequisite. However tailless mRNAs can be stabilized by tethering Pab1p (Poly-A binding protein) to the 3'UTR. See this paper.

Figure 4 of the linked paper.

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  Tethered Pab1p stabilizes mRNAs that do not receive a tail. (A)
  Strategy. The HDV self-cleaving ribozyme was inserted into the 3′ UTR
  of MFA2/MS2 RNA, yielding an mRNA without a poly(A) tail (A0) in vivo.
  (B) Oligo(dT) cellulose chromatography. RNAs were extracted from cells
  carrying the ribozyme-containing construct depicted inA, or the
  control plasmid used in previous figures. RNAs were fractionated by
  oligo(dT) cellulose chromatography, then analyzed by Northern
  blotting. The ribozyme-containing RNA is not retained by the column,
  whereas the control RNA is retained. (T) Total RNA prior to
  fractionation; (+) retained by the column; (−) not retained by the
  column. The mRNA runs slightly faster in the oligo(dT) retained (+)
  sample than in the total (T) RNA, because less RNA is loaded per lane.
  (C) S1 nuclease analysis. The 3′ end of mRNA prepared from a strain
  carrying the ribozyme-containing RNA was determined by S1 nuclease
  mapping. The position of undigested probe (65 nucleotides) and of the
  protected species (43 nucleotides) are shown. The prominent 3′
  terminus lies at the expected position for ribozyme cleavage. (D)
  Decay of ribozyme-cleaved RNA by a transcriptional pulse-chase
  experiment and Northern blotting. The turnover rate of
  ribozyme-cleaved reporter mRNAs was determined in strains with (left)
  or without (right) MS2/Pab1p. Time (in min) following transcriptional
  repression are given directly at top; half-lives are at bottom. (E)
  Quantitation of results. Amounts of mRNA were normalized to the level
  of 18S rRNA, at bottom of each lane in D. (•) MS2/Pab1p; (○) vector
  alone.


Prokaryotic UTRs do not have any general paradigm. To have a controlled inactivation of a mRNA in a prokaryotic system you can use riboswitches or  cis-repressive RNA (RBS complementary RNAs). RNA-guided RNA degradation is also possible in prokaryotes using the CRISPR-Cas system. See this paper. Proteins Cmr-1,2,3,4 &amp; 6 are required for this activity.