The paper focuses on monitoring and modelling of the cryptocurrency market. The application of the chosen research methods is based on the analysis of existing methods and tools of economic and mathematical modelling of time series research on the example of the cryptocurrency market. It is proved that the use of individual methods is not relevant, as they do not give an adequate assessment of the specified market, so a comprehensive approach is the most acceptable. Therefore, monitoring and modelling of some cryptocurrency pairs with different capitalization degree were implemented by fractal and recurrent methods of the financial markets. The daily values of currency pairs for the period from September 2015 to November 2019 were chosen as information basis for monitoring and modelling. The use of R/S modelling method make it possible to conclude the persistence of time series of the selected cryptocurrencies indicating that the market trends are clearly defined, the currency pair of XRP/USD has the highest level of trend resistance. To compare the obtained results, the comprehensive approach is offered using recurrent diagrams that help to determine the cryptocurrency stability. The results of modelling by the recurrent method show that the most stable cryptocurrencies are the ones with the highest capitalization, namely Bitcoin and Ripple.