Context. Dust grains are determinant for setting the chemical, physical, dynamical, and radiative properties of all the media in which they are present. Their influence depends on the grain composition, size, and geometrical structure which vary throughout the lifecycle of dust. Particularly grain growth arises in dense molecular clouds and protoplanetary disks as traced by an enhancement of the dust far-IR emissivity and by the effects of cloudshine and coreshine. Aims. Our aim is to investigate the imprint of the grain characteristics on the dust unpolarised optical properties from the visible to the far-IR wavelengths for isolated grains as well as for aggregates. Methods. Using optical constants for both carbonaceous and silicate materials, we derive the absorption and scattering efficiencies, the asymmetry factor of the phase function, the single scattering albedo, and the mass opacity for isolated grains and aggregates, using either the Mie theory or the Discrete Dipole Approximation (DDA). We investigate the effects of the size, porosity and shape of the grains, and of the monomers constituting the aggregates, on the optical properties. Besides, for aggregates we study the influence of the number of monomers and of mixing monomer sizes. Results. Grain structure changes result in optical property variations at all wavelengths. Porosity, grain elongation, as well as aggregation all produce an increase in the far-IR opacity. The spectral dependence of this increase depends on the nature of the material composing the grain: it is independent of the wavelength for insulators but not for conductors. In the case of aggregates, the far-IR increase does not depend on the monomer size and saturates for aggregates containing 6 or more monomers. In the visible and near-IR, the aggregate behaviour is reminiscent of a compact sphere of the same mass whereas at longer wavelengths, it is closer to the effect of porosity. Finally, for silicates, the mid-IR spectral feature at 18 µm is more sensitive to the details of the grain structure than the 10 µm feature. Conclusions. Dust optical properties, from the visible to the far-IR, are highly dependent upon the grain composition, size, and structure. This study provides a basis for understanding the range of variations achievable as a result of varying the grain characteristics. It emphasises the importance of considering the detailed grain structure in determining the dust optical properties and of using exact methods because approximate methods are unable to reproduce the entire range of the observed variations at all wavelengths.