Understanding the interplay between magnetism and superconductivity in nickelate systems is a key objective in condensed matter physics. Gaining microscopic insights into magnetism—particularly as it emerges near superconductivity—requires a synergistic approach that combines complementary experimental techniques with controlled tuning of external parameters. In this paper, we present a systematic investigation of the three-layer Ruddlesden-Popper (RP) nickelate ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$ using muon-spin rotation/relaxation ($\mu \mathrm{SR}$) and resistivity measurements. At ambient pressure, two incommensurate spin-density-wave (SDW) transitions are identified at $T_{\mathrm{SDW}}\simeq 132\mathrm{K}$ and $T^{*}\simeq 80–90\mathrm{K}$. Comparison of the observed internal magnetic fields with dipole-field calculations reveals a magnetic structure consistent with antiferromagnetically coupled SDW order on the outer two Ni-O layers, with smaller moments on the inner Ni-O layer. Above $T^{*}$, the moments lie primarily in the $ab$ plane, but below this temperature they undergo a subtle distortion and develop a $c$-axis component. The internal fields at the muon stopping sites appear abruptly at $T_{\mathrm{SDW}}$, suggesting a first-order-like nature of the SDW transition, which is closely linked to the charge-density wave (CDW) order occurring at the same temperature ($T_{\mathrm{SDW}}=T_{\mathrm{CDW}}$). Under applied pressure, all transition temperatures—including $T_{\mathrm{SDW}}, T^{*}$, and $T_{\mathrm{CDW}}$—are suppressed at a nearly uniform rate of $\simeq -13$ K/GPa. This behavior contrasts with that of the two-layer RP nickelate ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$, where pressure enhances the separation between the SDW and CDW transitions. The oxygen-isotope substitution (${}^{16}\mathrm{O} \to {}^{18}\mathrm{O}$) reveals that the CDW transition temperature shifts to higher values in the ${}^{18}\mathrm{O}$-substituted samples. The isotope effect on $T_{\mathrm{SDW}}$ and $T^{*}$ differs significantly. Specifically, when the CDW and SDW orders are intertwined, a notable isotope effect is observed on $T_{\mathrm{SDW}}$, leading to equal transition temperatures and nearly identical isotope shifts for both $T_{\mathrm{CDW}}$ and $T_{\mathrm{SDW}}$. In contrast, at $T^{*}$, where the SDW transition occurs independently of the CDW, no isotope effect is detected.

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• Magnetic order