The heterogeneity of grain quality can lead to limited predictability of qualitative and quantitative characteristics of the wheat yield, especially with an increase in the importance of drought and salinity caused by climate change. This study was undertaken with the aim of creating basic tools for phenotyping and assessing the sensitivity of genotypes to salt effects at the level of some wheat kernel attributes. The study considers 36 variants of the experiment, including four wheat cultivars—Zolotaya, Ulyanovskaya 105, Orenburgskaya 10, Orenburgskaya 23; three treatment variants—control (without salt) and two salts exposure (NaCl at a concentration of 1.1 g L−1 and Na2SO4 at a concentration of 0.4 g L−1); as well as three options for the arrangement of kernels in a simple spikelet—left, middle, and right. It has been established that the salt exposure had a positive effect on the percentage of kernel fulfilling in the cultivars Zolotaya, Ulyanovskaya 105, and Orenburgskaya 23 compared to control. The kernels of the Orenburgskaya 10 variety matured better in the experiment with Na2SO4 exposure, while the control variant and NaCl gave the same effect. When exposed to NaCl, significantly greater values of weight, transverse section area, and transverse section perimeter of the kernel were noted in the cv Zolotaya and Ulyanovskaya 105. Cv Orenburgskaya 10 responded positively to the use of Na2SO4. This salt caused an increase in the area, length, and width of the kernel. The fluctuating asymmetry of the left, middle, and right kernels in the spikelet was calculated. In the cv Orenburgskaya 23 the salts affected only the kernel perimeter among parameters examined. The indicators of the general (fluctuating) asymmetry were lower in the experiments with the use of salts, i.e., kernels were more symmetrical than in the control variant, both for the cultivar as a whole and when compared taking into account the kernel location in spikelet. However, this result was unexpected, since salt stress inhibited a number of morphological parameters: the number and average length of embryonic, adventitious, and nodal roots, flag leaf area, plant height, dry biomass accumulation, and plant productivity indicators. The study showed that low concentrations of salts can positively affect the fulfilling of kernels (the absence of a cavity inside the kernel) and the symmetry of the left and right sides of the kernel.