Jets, produced from the hard-scattered (high $p_{\rm T}$) partons in high-energy collisions, provide an important benchmark for perturbative quantum chromodynamics (pQCD) predictions. Measurements of the intra-jet properties in pp collisions provide a test of pQCD calculations and help to constrain the Monte Carlo models. Jet measurements in p$-$A collisions are sensitive to cold nuclear matter effects. Recent studies in small collision systems at high multiplicity exhibit signatures of collective effects that could be associated with hot and dense QCD matter known to be formed in heavy-ion collisions. In presence of such QCD matter the internal jet properties are also expected to be modified.
Therefore, these measurements in high-multiplicity pp and p$-$A collisions are important to establish whether deconfined QCD matter is indeed generated in such small systems. In this contribution, we report recent ALICE measurements of charged-particle jet properties, including mean charged-constituent multiplicity and fragmentation distribution for leading jets, in minimum bias pp collisions at $\sqrt{s}$ = 13$\,$TeV and minimum bias p$-$Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02$\,$TeV. In addition, the multiplicity dependence of these jet properties in pp collisions at $\sqrt{s}$ =13$\,$TeV is also presented. Results are compared to the predictions from various Monte Carlo generators.