Context. Wide-range spectral coverage of blazar-type active galactic nuclei is of paramount importance for understanding the particle acceleration mechanisms assumed to take place in their jets. The Major Atmospheric Gamma Imaging Cerenkov (MAGIC) telescope participated in three multiwavelength (MWL) campaigns, observing the blazar Markarian (Mkn) 421 during the nights of April 28 and 29, 2006, and June 14, 2006. Aims: We analyzed the corresponding MAGIC very-high energy observations during 9 nights from April 22 to 30, 2006 and on June 14, 2006. We inferred light curves with sub-day resolution and night-by-night energy spectra. Methods: MAGIC detects γ-rays by observing extended air showers in the atmosphere. The obtained air-shower images were analyzed using the standard MAGIC analysis chain. Results: A strong γ-ray signal was detected from Mkn 421 on all observation nights. The flux (E > 250 GeV) varied on night-by-night basis between (0.92±0.11) × 10-10 cm-2 s-1 (0.57 Crab units) and (3.21±0.15) × 10-10 cm-2 s-1 (2.0 Crab units) in April 2006. There is a clear indication for intra-night variability with a doubling time of 36±10_stat min on the night of April 29, 2006, establishing once more rapid flux variability for this object. For all individual nights γ-ray spectra could be inferred, with power-law indices ranging from 1.66 to 2.47. We did not find statistically significant correlations between the spectral index and the flux state for individual nights. During the June 2006 campaign, a flux substantially lower than the one measured by the Whipple 10-m telescope four days later was found. Using a log-parabolic power law fit we deduced for some data sets the location of the spectral peak in the very-high energy regime. Our results confirm the indications of rising peak energy with increasing flux, as expected in leptonic acceleration models.