Cirrus cloud formation is believed to be dominated by homogeneous freezing of supercooled liquid aerosols in many instances. Heterogeneous ice nuclei such as mineral dust, metallic, and soot particles, and some crystalline solids within partially soluble aerosols are suspected to modulate cirrus properties. Among those, the role of ubiquitous soot particles is perhaps the least understood. Because aviation is a major source of upper tropospheric soot particles, we put emphasis on ice formation in dispersing aircraft plumes. The effect of aircraft soot on cirrus formation in the absence of contrails is highly complex and depends on a wide array of emission and environmental parameters. We use a microphysical-chemical model predicting the formation of internally mixed, soot-containing particles up to two days after emission, and suggest two principal scenarios: high concentrations of original soot emissions could slightly increase the number of ice crystals; low concentrations of particles originating from coagulation of emitted soot with background aerosols could lead to a significant reduction in ice crystal number. Both scenarios assume soot particles to be moderate ice nuclei relative to cirrus formation by homogeneous freezing in the presence of few efficient dust ice nuclei. A critical discussion of laboratory experiments reveals that the ice nucleation efficiency of soot particles depends strongly on their source, and, by inference, on atmospheric aging processes. Mass and chemistry of soluble surface coatings appear to be crucial factors. Immersed soot particles tend to be poor ice nuclei, some bare ones nucleate ice at low supersaturations. However, a fundamental understanding of these studies is lacking, rendering extrapolations to atmospheric conditions speculative. In particular, we cannot yet decide which indirect aircraft effect scenario is more plausible, and options suggested to mitigate the problem remain uncertain.