The existence of a spheroid in M33 has been controversial for decades. While the galaxy does not appear to have a bulge, it definitely possesses a halo. I discuss these components in turn below.
3.1. M33 Bulge
M33 apparently does not possess a bulge in the classical sense of that term (Bothun 1992), although this has been the subject of debate (e.g. Minniti et al. 1993). Semantics aside, the galaxy hosts a small nucleus that can be fit with a bulge profile that dominates within ~ 0.1' of the galaxy center (Stephens & Frogel 2002). In a K vs. V - K color-magnitude diagram (CMD) of the brightest stars in the nucleus, Stephens & Frogel (2002) find young, intermediate-age, and old stars with a mean metallicity [Fe/H] = -0.26. It is not possible to obtain photometry of the low-mass MS stars in the nucleus with any observatory in operation or in development, so the constraints on the star formation history will be poor for the foreseeable future.
3.2. M33 Halo
The existence of a halo in M33 was also controversial in the past, such that the galaxy was sometimes referred to as a "pure disk" galaxy. However, in recent years a preponderance of evidence demonstrated beyond any doubt that M33 hosts a stellar halo. Although some regions of the disk have Hubble Space Telescope (HST) photometry reaching the stellar MS (Program 10190; PI Garnett), the halo-dominated regions beyond the disk have not been imaged at sufficient depth, so the star formation history of the M33 halo remains poorly constrained.
Chandar et al. (2002) kinematically segregated the stellar clusters of M33 into disk and halo components. Unresolved photometry and spectroscopy of the clusters enabled a rough age estimate that is prone to degeneracies between blue MS stars and blue HB stars. The young clusters in the sample have motions consistent with disk membership, but the old clusters show a much larger velocity dispersion, implying that 85% belong to the halo and 15% belong to the disk (Figure 1). The clusters with halo kinematics have a spread in relative ages of 5-7 Gyr, much greater than the globular cluster system in the Milky Way halo. Subsequent photometry of one globular cluster in the sample was sufficiently deep to demonstrate that its blue spectrum is due to the presence of blue MS stars and not blue HB stars, proving its age is 5-8 Gyr than the other globular clusters in their sample, but unfortunately the cluster cannot be kinematically assigned to either the disk or halo (Chandar et al. 2006).
Figure 1. Difference between local disk velocity and measured cluster velocity as a function age in M33 clusters (Chandar et al. 2002), where cluster age is estimated from integrated photometry and spectroscopy. The spread in age for globular clusters in the M33 halo is much larger than it is for such clusters in the Galactic halo. Plot provided courtesy of R. Chandar.
(Sarajedini et al. (2006)) found evidence for a halo using 64 RR Lyrae variables they identified in the galaxy. They estimated reddenings from their minimum V - I colors and metallicities from their periods. The resulting distributions of reddening and metallicity imply they belong to two distinct populations, associated with the disk and halo, providing evidence for these components in the field that complements the evidence in clusters (above). RR Lyrae stars are only present in ancient (> 10 Gyr) populations, implying that both the M33 disk and halo host at least some ancient stars. In the [Fe/H] distribution for the halo RR Lyrae stars, the peak is at -1.3, although the metallicities of the RR Lyrae stars are not necessarily representative of the underlying population, because metallicity affects the HB temperature distribution and thus the fraction of HB stars falling in the instability strip.
McConnachie et al. (2006) also found evidence for a halo field population. From spectroscopy of 280 stars, they were able to segregate their sample into three components using both kinematics and metallicity: the halo, the disk, and a third unknown component possibly associated with a stellar stream. The halo component exhibits a mean [Fe/H] of -1.5 and a velocity dispersion of ~ 50 km s-1, but the data provide no age constraints.
Using star counts of RGB and AGB stars along the minor axis, Teig (2008) found a break in the surface brightness profile at 11 kpc, where the profile changes from that of an exponential disk to a power-law halo (see also Teig et al. 2009). The CMD of these bright stars implies the halo population is dominated by stars older than 3 Gyr with a mean [Fe/H] of -1.2. This is in good agreement with the CMD of Brooks et al. (2004), who obtained V and I photometry in the halo outskirts and found a mean [Fe/H] of -1.24.
The next logical step in the investigation of the M33 halo is deep photometry reaching its low-mass MS stars. This is the only way to unambiguously characterize the age distribution in the field population. While such photometry has been obtained in multiple regions of the M31 halo with HST (discussed below), the M33 halo remains unexplored, with very loose constraints on its star formation history.