The following are cellular automata rules (in standard semitotalistic rule notation) which "seem like" they should have moving patterns (or for which it would be especially interesting to find such patterns), but for which my search software has been unsuccessful:

B35/S234578"land rush" |
Chaotic growth evolves into stable patchwork of horizontally and vertically plowed regions. Has various small oscillators including Life's clock, and arbitrarily high period oscillators via a simulation of the one-dimensional B01/S02 automaton (e.g. between two out-of-phase rows of farmland). Ok, maybe there's no reason for it to have gliders but it's one of the more interesting rules I've seen. The stable regions can have small oscillators in them, maybe there are gliders and more complicated patterns if we're given an initial background of farmland rather than blank space. |

B345/S5"longlife" |
Complicated long-lived oscillating patterns of alternating rows seem to be trapped inside a bounding diamond. Are they really unable to grow? |

B01345/S01234 |
Most random initial configurations quickly decay into small oscillating blobs, mostly with period two. If the initial density is around 59%, the phase ambiguity leads to much larger intertwined black and white blobs with stable boundaries edged by higher period oscillators. |

B017/S1 |
Random fields at 50% turn into grids of 4c/8 and 7c/14 replicators blocked between small black blobs. Random fields at 25% turn into blobby unmoving mixtures of black and white areas. Very sparse random fields form grids of replicators again, gradually accumulating blobs where the replicators interact until all replicators are blocked between pairs of blobs. Except for the replicators, random patterns become static very quickly. |

B02/S |
Random fields at 25% turn into long-lasting blobs which decay into small low-period oscillators. Higher density random fields form a single large blob. |

B3/S23456 |
An interesting mixture of slow chaotic accretion and horizontal and vertical shoots. When a chaotic blob meets a shoot, it can occasionally form a "runner" that grows along the shoot; when the runner catches up to the head of the shoot it starts a new blob of chaos. Also supports a few small oscillators including Life's clock. Very similar to B3/S012345678 "Life without death" which of course has no spaceships. |

B3/S45678 |
A very slowly growing black blob filled with intricate tree-like patterns of white dots. Has small oscillators of periods 3, 5, 6, 8, and 22. Searched unsuccessfully with gfind to level L128. |

B34/S03456 |
One of the rules studied by Wolfram. An expanding octagonal region filled with stable junk. Has many small p2 and p4 oscillators. |

B345/S456 |
Extremely slowly growing black diamonds mostly filled with a stable bacterial matrix but with some long-term chaotic fluctuations. One of two narrow connections between the B35xx/S45xx and B34xx/S4xx blocks of rules. |

B345/S4567 |
Stable black diamonds filled with white dots. |

B34567/S0678 |
Slowly growing mazes. Symmetric under black/white reversal. |

B34567/S348 |
The only B3xx/S348 rule without known spaceships. Slightly rounded diamond-shaped growing chaos. |

B345678/S15678 |
Stable filled octagons. |

B345678/S014578 |
Dense expanding chaos. The only B3xx/S014578 rule for which no gliders are known. |

B345678/S458 |
Dense chaos fills an expanding diamond-shaped region. |

B345678/S47and B345678/S478 |
Slowly expanding chaos. |

B348/S4 |
Quickly decays to isolated 4-cell p2 oscillators. |

B34678/S23578 |
Expanding chaos. |

B3478/S24568 |
Expanding chaos. |

B35/S46 |
Has p2 and p4 oscillators, but most patterns quickly decay to nothing. One of two narrow connections between the B35xx/S45xx and B34xx/S4xx blocks of rules. |

B36/S1567 |
Quickly stabilizes to isolated domino still lifes and low-period oscillators. Has arbitrarily high-period oscillators formed out of 2x2 blocks of cells. |

Even more than gliders for these rules, I would like a mathematical proof that certain rules (e.g. rules with B23, with B3 and none of S0-S5, or with all of B3/S34567) are unable to support gliders.